Light and Matter Id / Licht und Materie Id

ENCYCLO PEDIA OF PHYSICS

EDITOR IN CHIEF

S. FLUGGE

VOLUME XXV/2d

LIGHT AND MATTER Id

BY

H. BILZ . D. STRAUCH· R.K. WEHNER

EDITOR

L. GENZEL

WITH 139 FIGURES

SPRINGER-VERLAG BERLIN HEIDELBERG NEW YORK TOKYO

1984

HANDBUCH DER PHYSIK

HERAUSGEGEBEN VON

S. FLUGGE

BAND XXVj2d

LICHT UND MATERIE Id

VON

H. BILZ . D. STRAUCH· R.K. WEHNER

BANDHERAUSGEBER

L. GENZEL

MIT 139 FIGUREN

SPRINGER-VERLAG BERLIN· HEIDELBERG· NEW YORK· TOKYO

1984

Professor Dr. SIEGFRIED FLUGGE

Physikalisches Institut der Universitiit, D-7800 Freiburg i. Br.

Professor Dr. LUDWIG GENZEL

Max-Planck-Institut fUr Festkorperforschung, D-7000 Stuttgart

ISBN-13: 978-3-642-46435-5 e-ISBN-13: 978-3-642-46433-1 DOl: 10.1007/978-3-642-46433-1

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Preface

The dynamical properties of solids have recently attracted renewed interest in connection with the increasing understanding of phase transitions and related phenomena. In particular, soft modes or, more generally, phonon 'anomalies' seem to play an important role in structural and electronic phase transitions, such as ferroelectric or superconducting transitions. The understanding of the mechanisms responsible for the occurrence of unusually low frequencies in phonon spectra requires a detailed analysis of the microscopic forces governing the lattice vibrations. Of particular importance is the influence of the electronlattice interaction in the adiabatic approximation which in many cases is the origin of peculiarities in the phonon self-energy.

In this work the vibrational spectra of pure non-metals and of those containing point defects are investigated.' In these materials the interrelation between the pseudo-harmonic forces (determining the phonon dispersion relations) and the non-linear anharmonic and electron-phonon forces (as they act in infrared and Raman spectra) is most obvious and can be quantitatively analysed in terms of appropriate models. The main task is to arrive at a physically correct treatment of electronic degrees of freedom, as for example in an electronic 'shell' model, which leads to the description of phonon spectra in terms of long-range polarizabilities and short-range deformabilities. The purpose of our review is to stimulate further investigations which, we hope, will result in explicit relations between the parameters of the semi-microscopic models and the matrix elements from the electronic band structure. Our contribution is restricted to vibrational spectra to emphasize the 'phonon' aspects of infrared absorption and Raman spectra. Therefore, resonant Raman spectra and related phenomena are bareleey discussed, and the reader is referred to the rich literature in these fields. The same holds for an explicit analysis of dynamical aspects of phase transitions and of the interesting Raman spectra of superconductors and valence-mixing crystals. The authors hope that their review provides a coherent presentation of the basic concepts useful for an understanding of the dynamical properties of solids as they manifest themselves in their vibrational spectra.

The authors wish to thank many colleagues and co-workers for stimulating discussions, helpful criticism and substantial support during the work on the manuscript.

In particular, they would like to mention some specific contributions to this article. The essential contents of Sect. 22b on the lattice relaxation are parts of unpublished results by J.B. Page; the permission to publish them here is gratefully acknowledged. T.P. Martin kindly agreed to contribute a survey on finite

VI Preface

crystals (Sect. 32) while E. Kiefer-Schroder was kind enough to prepare the tables of elastic and dielectric constants (Sect. 39).

In addition, the authors are grateful for instructive comments and helpful suggestions by L. Genzel, R. Klein, F.W. de Wette, B. Gliss, W. Kress, D. Smith, W. Weber and R. Zeyher. The final version of the manuscript benefitted from a critical reading of parts of the manuscript, concerning physical and linguistic aspects, by M. Buchanan, R.J. Bell, W. Kleppmann, W. Kress and T.P. Martin. We also thank Mrs. Eva Genzel very much for the preparation of the subject index. Finally the untiring help of our secretaries Mrs. R. Ocal, Mrs. E. Brtigmann, Mrs. A. Wiilti and Mrs. G.!. Keck in preparing the different versions of the manuscript was essential for the completion of the article.

It is a great pleasure to thank the editors and the publishers, in particular L. Genzel and H. Mayer-Kaupp, H. Lotsch, K. Koch and K.-H. Winter for their remarkable patience and co-operation during the time of the production of the article.

H. BILZ, D. STRAUCH, and R.K. WEHNER

Contents

Vibrational Infrared and Raman Spectra of Non-Metals By H. BILZ, Max-Planck-Institut flir Festkorperforschung,D-7000 Stuttgart, Fed. Rep. of Germany, D. STRAUCH, Institut flir Theoretische Physik der Universitat Regensburg, D-8400 Regensburg, Fed. Rep. of Germany, and R.K. WEHNER, Fachbereich Physik der Universitat Munster, D-4400 Munster, Fed. Rep. of Germany

A. Introduction . . . .

1. Historical survey.

2. Outline of the theory of infrared absorption and Raman scattering . . . . .

a) Macroscopic aspects. b) Microscopic aspects .

B. Phonons in insulators . . .

3. General properties of phonons . a) Dynamic and thermodynamic stability of solids b) The adiabatic approximation . . c) Force constants. . . . . . . . . . d) Symmetry properties of phonons . . e) The pseudo-harmonic approximation

4. Ionic crystals . . . . . a) The rigid-ion model. . . b) Dipole models . . . . . c) The breathing shell model d) Ionic deformabilities. . . e) Non-central and many-body forces and the elastic

properties of crystals. . . . . . . . . . . .

5. Covalent crystals . . . . . . . . . . . . . . a) Formal force constants and general properties b) Dipole models . . . c) Bond-charge models. . . . . . . . . . .. d) Valence force fields . . . . . . . . . . . . e) Crystals of partially ionic and partially covalent

character. . . . . . . . . f) Sum rule of lattice vibrations. . . . . . . . .

1

2

4 4 5

9

9 10 15 16 18 20

21 21 26 29 31

38

44 44 44 46 48

49 50

VIII Contents

6. Microscopic theory, models, and macroscopic quantities 51 a) Overlap theory . . . . . . . . . . . 52 b) The dielectric function method . . . . . . . 54 c) The direct 'frozen-in' phonon approach . . . 57 d) Charges and polarizabilities of ions and bonds 58 e) Electric fields and effective charges in ionic solids 61 f) Fields and charges in covalent solids . . . . . 70 g) The microscopic description of charges and fields 74

C. Interaction of photons with matter . . . . . . . . 77

7. Theory of interaction of photons with particles 77 a) Non-relativistic theory of inelastic scattering 77 b) Gauge invariance in electromagnetic interaction. 82 c) Dielectric constant of electrons . . . . . . . 86 d) Light scattering by electrons . . . . . . . . 90 e) Interaction of photons with electrons and ions 91 f) Polaritons in the harmonic approximation 92

8. Infrared absorption and dielectric response . . . 95 a) Dielectric susceptibility . . . . . . . . . . 96 b) Absorption of radiation (fluctuation-dissipation theorem) . 99 c) Frequence-dependence and thermodynamic definitions

of the susceptibility, sum rules 101 d) Static susceptibility . . 103

9. Raman scattering of light. . . . 108 a) Introduction . . . . . . . . 108 b) Quantum theory of spontaneous Raman scattering 109 c) Adiabatic representation. . . . . . . . . 112 d) Polarizability theory. . . . . . . . . . . 115 e) Green function theory of Raman scattering. 116 f) The 0)4 law. . . . . . . . . . . 120 g) Polariton picture of light scattering . . . . 123 h) Resonant Raman scattering (RRS). . . . . 124 i) Rayleigh, Brillouin, and Hyper-Raman scattering 125

D. Expansion theory of susceptibilities and polarizabilities 125

10. General lattice potential . . . . . . . . . . . 126 a) The undeformed lattice . . . . . . . . . . 126 b) The lattice in a static electric field and under deformation 127

11. Lattice dipole moment . . . . . . . . . . . . . . . . . 132 a) The undeformed lattice . . . . . . . . . . . . . . . 132 b) The lattice in a static electric field and under deformation 132

12. Lattice and electronic susceptibility 133 a) Formal expansion of the susceptibility 133 b) The harmonic approximation. 134 c) Anharmonic susceptibility . . . . . 137

Contents IX

. d) The anharmonic dispersion oscillator 138 e) The damping function. . . . . . . 140 f) The renormalized dipole moment . . 141 g) The general form of the lattice susceptibility 141 h) Coupling of dispersion oscillators . . . . . 142 i) Anharmonic coupling parameters . . . . . 142 j) The susceptibility under external pressure and

in a static field . . . . . . . . . . . . . . 142

13. Lattice polarizability and Raman scattering . . . 143 a) Formal expansion of the electronic susceptibility 143 b) Harmonic approximation . . . . 144 c) Anharmonic treatment. . . . . . 145 d) Raman scattering in cubic crystals. 145 e) Raman coupling parameters . . . 146 f) Effects of static fields and external pressure. 147

E. Interpretation of experimental spectra . . . . . . . . . . . 148

14. Model theory of infrared absorption and Raman scattering 148 a) General features of infrared and Raman processes. . 148 b) Microscopic and model treatment of electron-phonon

interaction . . . . . . . . . . . . . . . . . . . 151 c) Shell model treatment of Raman scattering. . . . . 153 d) Bond charge and bond polarizability in infrared and

Raman processes . . . . . . . . . . . . 156

15. Infrared spectra of ionic crystals. . . . . . . . . . . 157 a) Qualitative classification of infrared spectra. . . . . 157 b) The infrared spectra of alkali halides: anharmonic effects. 163 c) Critical point analysis . . . . . . . . . . . 167 d) Density of states approximation. . . . . . . 169 e) The effect of short-range cubic anharmonicity . 171 f) The effect of quartic and higher anharmonicity 172 g) Coulomb anharmonicity . . . . . 180 h) Absorption at very low frequencies 183 i) Non-linear dipole moments. . . . 186 j) The effect of ionic polarizability. . 190 k) Final states interactions of phonons: anharmonic

broadening and bound states . . . . . . . . . 190 1) Line widths of dispersion oscillators and temperature-

dependence. . . . . . . . . . . . . . . . . 191 m) Discussion of other diatomic ionic crystals . . . 198 n) Cubic crystals with three and more ions in a cell 203

16. Infrared spectra of covalent crystals . . . . 208 a) General features of the spectra . . . . . . 208 b) Spectra of crystals with diamond structure . 209 c) Covalent crystals with linear dipole moments 214

x Contents

17. Infrared spectra of crystals with mixed ionic and covalent character. . . . . . . . . . . . . . . . . . . . . 214

a) The concurrence of anharmonicity and non-linear dipole moments. . . . . . . . . . . . . . . . 214

b) Spectra of crystals with zincblende structure 216 c) Spectra of perovskites . . . . . . . 219 d) Spectra of low-symmetry crystals . . 224 e) Spectra of amorphous semiconductors 227

18. Raman scattering from ionic crystals. . 229 a) Raman spectra of cubic ionic crystals 230 b) Other diatomic ionic crystals 244 c) Perovskites. . . . . . . . . . . . 245 d) Other ionic crystals . . . . . . . . 246 e) Photoelasticity and Raman scattering 246 f) First-order Raman scattering. . . . 251

19. Raman spectra of covalent and partially ionic crystals 252 a) Spectra of diamond and its homologues 252 b) Spectra of III-V and II-VI compounds. 258

F. Lattices with point defects . . . . . 262 20. Types of defects and their effects . 262

a) Introductory remarks . . . . 262 b) Point defects, vacancies . . . 264 c) Defect-induced infrared and Raman spectra. 266 d) Localized modes, gap modes . . . . . . . 267 e) Resonant modes . . . . . . . . . . . . 271 f) Off-center and molecular defects: Tunnelling motion. 280 g) Internal vibrations of molecular defects . . . . 283 h) Interstitials. . . . . . . . . . . . . . . . . 286 i) Effects of defect clusters and defect concentration 288 j) Dislocations, surfaces . . . . . . . . . . 289

21. Information contained in defect-induced spectra . 289 22. Lattice dynamics of impure lattices. . . . . . . 295

a) Introduction: Molecular model - the nature of perturbations due to a defect . . . . . . . 295

b) Lattice distortions - method of lattice statics 297 c) Equation of motion of the perturbed lattice. 300 d) Symmetry considerations. . . . . . . . . 303 e) Lifshitz method for the solution of the equation of motion

- localization of perturbations . . . . . . . . . 309

23. The Green function of the harmonic perturbed lattice. 315 a) Real Green function and T matrix. . . . . 315 b) The complex Green function . . . . . . . . . . 317 c) Resonances: Localized and resonant modes. . . . 320 d) Eigenvalue treatment of the Green function and T matrix

in the impurity space . . . . . . . . . . . . . . . . 323

Contents XI

24. Properties of the perturbed harmonic lattice Green function . 324 a) Kramers-Kronig transform. . . . . . . . . . . . . . 324 b) Normalization of the perturbed resonance-mode eigenvectors:

An effective mass of the resonance vibration . . . . . .. 325 c) Approximate form of the Green function and of the T matrix

near a resonance frequency: Width and intensity . . . .. 326

25. Applications of Green functions: Phonon spectra in perturbed crystals . . . . 328 a) Phonon density of states 328 b) Dielectric susceptibility 331 c) Raman scattering . . . 341 d) Resonance Raman scattering 347

26. Dynamics of lattices with interstitial or molecular defects 351 a) Formulation of the problem . . . . . . . . . 351 b) Standard procedure - application to interstitials. . . 352 c) Formalism modified for molecular defects . . . . . 355

27. Shell-model treatment of the dynamics of perturbed lattices and the model theory of infrared-absorption and Raman-scattering spectra . . . . 358 a) The use of shell models . . . . . . . . . . 358 b) Effective force constants . . . . . . . . . . 361 c) Shell-model extension of the Lifshitz formalism 363 d) Shell-model interpretation of the effective charge 373 e) The higher-order dipole moments . . . . . . . 376 f) Shell-model interpretation of the Raman scattering intensity 377

28. Concentration effects. . . . . . . . . . . . . . . . . 382 a) Introduction: Diagrammatic expansion . . . . . . . 382 b) Low-concentration single-site scattering approximation. 386 c) Self-consistent approximation. . . . . . 388 d) Coherent-potential approximation (CPA) . 388 e) Applications . . . . . . . . 389

29. Mixed crystals. . . . . . . . . 394 a) One- and two-mode behaviour 394 b) Theoretical models . . . . . 399 c) Changes in the lattice constant and Ivey relation 405

30. Anharmonic effects in perturbed crystals . . . . . 406 a) Introduction: Resonance modes in analogy to the Rest-

strahlen or Raman oscillator . . . . . . . . . 406 b) Qualitative aspects of the anharmonic self-energy

in perturbed crystals. . . . . . . . . . . . . 408 c) Diagonal and off-diagonal elements of the perturbed

self-energy . . . . . . . . . . . . . . . . . . . 409 d) Low-order contributions to the self-energy . . . . . 413 e) Approximate form of the anharmonic Green function 416

XII Contents

f) Intensity of resonances. . . . . . . . . . . . 418 g) Anharmonic shift of resonance-mode frequencies 424 h) Isotope effects . . . . . . . . . . . . . 428 i) Anharmonic width of resonances . . . . . 431 j) Multi-phonon spectra, sidebands, overtones. 434 k) Higher-order effects . . . . . . . . . . . 442

31. Phonon frequency shift from bulk and local strain due to temperature variation, pressure, and lattice distortion in defective crystals. . . . . . . . . . 445 a) Equilibrium positions . . . . . . . . . . 445 b) Low-concentration approximation. . . . . 451 c) Relation between lattice and elasticity theory 453 d) Static distortions . . . . . . . ·460 e) Pressure-induced frequency shift 463 f) Thermal expansion . . . . . 466

32. Finite crystals. By T.P. MARTIN. . 467 a) Introduction . . . . . . . . . 467 b) Continuum theory of finite crystals 467 c) Lattice dynamics of finite crystals . 471

33. Present and future problems in lattices with defects. 473

G. Dynamical theory of interacting phonon systems. 474

34. Basic concepts. . . . . . . . . . . . . . 474 a) Introductory comments . . . . . . . . 474 b) Normal coordinates and lattice Hamiltonian 476 c) Equilibrium correlation and Green functions 479 d) Double-time Green functions: Harmonic approximation 481 e) Double-time Green functions: Spectral representations. 483

35. Functional methods . . . . . . . . . . . . .'. . 487 a) Non-equilibrium Green functions . . . . . . . . 487 b) Generalized thermodynamic potentials and cluster

expansion . . . . . . . 491

36. Phonon dynamics . . . . . 494 a) Basic equations of motion 494 b) Equilibrium positions . . 497 c) The renormalized harmonic approximation. 500 d) Dyson equation with dispersive interactions 504

37. Vertex renormalization. . . . . . . 509 a) Vertex part integral equations. . . 509 b) Self-energy with vertex corrections. 516

38. Simple approximations and results. . 524 a) Self-energy and retarded Green functions. 524 b) Free energy . . . . . . . . . . . . . 530

Contents

H. Appendices. . . . . . . . .

39. Tables of elastic and dielectric constants

40. Tables of selection rules, etc.

References. .

Subject Index

XIII

535

535

550

557

593

Vibrational Infrared and Raman Spectra of Non-Metals

By

H. BILZ, D. STRAUCH, and R.K. WEHNER

With 158 Figures

A. Introduction

The vibrational properties of crystals determine the photon infrared absorption, inelastic neutron scattering and, to a large extent, inelastic photon scattering by phonons, i.e., Raman scattering. The interpretation of infrared and Raman spectra requires, therefore, an understanding of the basic features of lattice dynamics. The quantum theory of solids describes the crystal properties in terms of elementary excitations and their mutual interactions. Dynamic properties are represented by phonons (lattice vibrations) and their interactions mainly with other phonons (anharmonicity), electrons (electron-phonon coupling), and photons (interaction with radiation). This characterizes the scope of the article. Its emphasis is on the interrelation between theory and experiment, i.e., on the microscopic or model interpretation of experimental spectra.

Work on infrared absorption and Raman scattering of non-metallic solids was summarized twenty years ago in two review articles in this Encyclopedia by LECOMTE (1958) and by MIZUSHIMA (1958). Since then, many important new experimental data have become available, thanks to very refined techniques for measuring spectra of pure and imperfect single crystals. Furthermore, the theory of the interactions of phonons and photons has been much improved, stimulated by the remarkable development of the theory of lattice vibrations and the mathematical techniques of many-body physics.

In this article we describe the dispersion and absorption of infrared radiation and Raman scattering in non-metallic crystals, both perfect and containing point defects. At present, quantitative calculations are usually restricted to diatomic crystals, especially those with simple structures such as the alkali halides or germanium and its homologues, since our knowledge of lattice vibrations is still rather poor for polyatomic and low-symmetry crystals.

2 Introduction Sect 1

The theory of lattice vibrations, as reviewed by COCHRAN and COWLEY (1967) in Vol. XXV /2a of this Encyclopedia, constitutes the background and the natural starting point for our investigations. There are other recent reviews by LUDWIG (1967), COCHRAN (1971), MARADUDIN et al. (1971), SINHA (1973), and several articles in: HORTON and MARADUDIN (eds., 1974). A summary of the developments during the last few years is given in Chap. B.

An important part of the theory of lattice vibrations is the construction of models. A good example is the so-called shell model for phonons (see Sect. 4) which describes the adiabatic linear electron-ion interaction in terms of localized charges and coupling constants. This provides a natural explanation of some long-range ion-ion forces in insulators in terms of induced dipole forces. Anharmonic extensions of this shell model and its modifications seem very desirable, and first attempts in this direction are discussed.

Models can often give a qualitatively and sometimes quantitatively correct description of certain processes in terms of a few parameters. They provide, therefore, an orientation for microscopic approaches. Furthermore, the parameters of workable models often show systematic trends when classes of similar crystals are compared. From this we can derive certain "rules" or phenomenological concepts which are a familiar aspect of chemistry and physics. Because in physics one undertakes the study of increasingly complex systems, such models and rules are indispensable, even if only as temporary tools. This will become obvious during the progress of discussion in the present article.

Our contribution is complementary to that of BIRMAN (1974) in Vol. XXV /2 b of this Encyclopedia. BIRMAN emphasizes the symmetry-related properties of crystals interacting with a radiation field (e.g., selection rules). We focus attention on the dynamic aspects: for example, the relative importance of cubic and quartic anharmonic coupling of phonons in certain absorption processes. We frequently quote BIRMAN'S results on the symmetry properties of phonons in our presentation of the subject.

1. Historical survey. Around 1800, F.W. HERSCHEL observed infrared radiation for the first time with the help of a thermometer. The first systematic investigations were done almost a century later by H. RUBENS and his coworkers (RUBENS, 1900) not long after the discovery of the Reststrahlen by NICHOLS (1897).

EINSTEIN (1907) was the first to analyze the lattice specific heat in terms of a single representative lattice mode ("Einstein"-oscillator). In 1909 MADELUNG published a paper "Molekulare Eigenschwingungen" in which he established a connection between the long-wavelength infrared vibration frequencies and the elastic constants of diatomic cubic crystals. His paper (MADELUNG, 1909; see also SUTHERLAND, 1910; EINSTEIN, 1910; and MADELUNG, 191Oa, b) marks the beginning of the atomistic theory of lattice dynamics, which occurred three years before experimental evidence of crystal structures had been found by VON LAUE (1912). In 1912, shortly after DEBYE had treated the problem of the specific heat of crystals in the continuum approximation, BORN and VON KARMAN (1912) wrote their paper on the normal vibrations of three-dimensional lattices. With

Sect. 1 Historical survey 3

this work, the basic concepts of the theory of lattice vibrations were established 1.

The problem of dispersion and absorption of electromagnetic radiation in crystals was long considered within the framework of DRUDE'S theory (1900), which describes the damping of infrared-active lattice modes by an ad hoc damping constant. However, the discovery of two or more maxima in some Reststrahlen bands of alkali halides by RUBENS and his co-workers (see the bibliography by PALIK, 1960) implied, as early as 1910, the existence of a more complicated mechanism. The possibility was discussed that these maxima might be produced by frequency-dependent damping (MADELUNG, 191Ob).

A theory of damping requires consideration of the anharmonic part of the lattice potential. The first discussion of a linear chain was given by PAULI (1925). A few years later PEIERLS (1929) developed a general theory of irreversible processes in crystals and found the "Umklapp-Prozesse" to be responsible for the finite thermal conductivity of crystals. Building upon the ideas of PAULI and PEIERLS, BORN and BLACKMAN (1933), BLACKMAN (1933) investigated the infrared absorption of crystals due to the anharmonic coupling of lattice vibrations, and explained qualitatively the structure of the absorption and reflection spectra found by CZERNY (1930), BARNES (1931, 1932), and KORTH (1932) in some alkali halides.

There was a distinct halt in the development of the theory in 1933. Two papers, by BARNES et al. (1935) and BLACKMAN (1937) did not raise much interest among the experimentalists. The book by BORN and HUANG on lattice dynamics appeared in 1954, and in 1955, the paper by LAX and BURSTEIN on infrared absorption in crystals was published. Both stimulated further investigations. The first calculations of damping functions of some alkali halides were carried out in 1960 (MITSKEVICH, 1961; BILZ et aI., 1960; BILZ and GENZEL, 1962). After the perturbation treatment by SZIGETI (1960), the first dispersion formulas which correctly included the resonance absorption were obtained from the equation-of-motion method. NEUBERGER and HATCHER (1961) discussed the classical high-temperature limit, while VINOGRADOV (1963) investigated the general case. Further results by MARADUDIN and FEIN (1962), MARADUDIN and WALLIS (1962) and by COWLEY (1963) were obtained with the help of thermodynamic Green functions. A very compact formulation of the theory was produced by WEHNER (1966, 1967), using functional methods. To date, however, only the infrared spectra of a few simple ionic and homopolar crystals have been analyzed quantitatively in terms of parameters from the anharmonic potential and the non-linear dipole moment (see Sect. 15).

The history of the inelastic scattering of light by phonons (Raman scattering) shows a similar development. After LOMMEL's (1871) and SMEKAL's (1923) early discussion of inelastic scattering of light, the Raman effect in crystals was detected by LANDSBERG and MANDELSTAM (1928) at about the same time as RAMAN and KRISHNAN (1928) found the effect in liquids and gases. A few years later FERMI and RASETTI (1931) discovered the second-order Raman effect

1 For further details of the situation around 1912 see the historical remarks by BORN and DEBYE

at the Copenhagen Conference 1963 (BORN, 1963; DEBYE, 1963).

4 Introduction Sect. 2

which has a strong relation to the two-phonon infrared absorption. The classical description of the first-order effect was published by MANDELST AM et al. (1930); and the quantum theory was presented by T AMM (1930) together with his theory on Brillouin scattering. After PLACZEK'S excellent review (1934), there was again a lapse of many years before further investigation of the Raman effect was stimulated by BORN and BRADBURN'S paper (1947) on the Raman effect in second order and SMITH's (1948) subsequent attempt to analyze the diamond spectra. In the following years, parallel with the improved understanding of lattice dynamics, the theory has been refined using Green function methods and satisfactory models for the lattice vibrations (COWLEY, 1963; LOUDON, 1964).

The development of laser techniques extended the field of possible Raman scattering experiments. The present state of affairs (WRIGHT, 1969; BALKANSKI, 1971; ANDERSON, 1971, 1973; BIRMAN, 1974; CARDONA, 1975; HAYES and LOUDON 1978) corresponds to the situation in infrared physics, although it must be kept in mind that the theory of the Raman effect is basically more complicated than the theory of dielectric susceptibility. The rapid development of infrared and Raman physics, in particular the theoretical analysis of the spectra, would have been impossible without a parallel progress of inelastic neutron scattering. After a first decade of structural investigations by neutron scattering which began in 1946, the technique of inelastic scattering was developed mainly by BROCKHOUSE and his co-workers (refer to: BROCKHOUSE, 1961; EGELSTAFF, 1965). There now exists a great collection of neutron data on phonons in metals and non-metals which provides an indispensable basis for any further analysis of dynamic properties. For further details the reader is referred to recent reviews by DOLLING (1973, 1974), to the bibliography by LAROSE and VANDERWAL (1974), and to the Japanese J.A.E.R.I.-M.Report (1976). A collection of data for insulating solids is given in the phonon atlas of BILZ and KRESS (1979).

2. Outline of the theory of infrared absorption and Raman scattering.

a ) Macroscopic aspects. In the macroscopic theory of infrared absorption and Raman scattering, the interaction of external radiation with a crystalline specimen is described by macroscopic quantities such as reflection, transmission, and absorption coefficients. From these quantities, which depend on the shape of the crystal and the orientation and polarization of the incident light, the bulk optical constants are obtained after some (frequently quite lengthy) calculations.

The technique of these calculations is discussed in the article by BELL (1967) in Vol. XXV /2a of this Encyclopedia. The result is most conveniently expressed by the shape-independent dielectric constant e which connects the total macroscopic field E in a crystal with the induced polarization P (see, for example, SZIGETI, 1971),

D:=E+4nP=eE. (2.1)

In this description the crystal is completely characterized by the complex secondrank tensor e = 8' + i e" which represents the response function of a crystal to an electric field. In the following, we assume that e itself does not depend on the field, i.e., e is a linear response function.

Sect. 2 Outline of the theory of infrared absorption and Raman scattering 5

The causal behavior of the crystal as a linear system leads to relationships between the real and the imaginary parts of e. These so-called Kramers-Kronig relations hold in a similar form for the other pairs of optical constants nand k, Rand cp. An extensive discussion is given in the article by BELL (1967). The Kramers-Kronig relations are very useful for the derivation of the whole set of optical constants if only one of them, for example the reflectivity, R, is known with sufficient accuracy for a large spectral region (see Chap. E).

In a similar way, Raman inelastic scattering of light is described macroscopically by the cross section for the scattered light per unit solid angle (see BORN and HUANG, 1954 and Eq. (9.16)), given by:

(2.2)

Here Wi,s is the frequency, e~s is the directional cosine of the field strength (i = incident, s = scattered). The differential scattering cross section (2.2) includes the fourth-rank Raman tensor i of the crystal (Sect. 9).

The macroscopic theory of Raman scattering has been discussed by BORN and HUANG (1954) and, more recently, by CARDONA (1975) and by HAYES and LOUDON (1978). For the dielectric constant e and the Raman scattering tensor i, the symmetry of the crystal is important. For example, in a cubic crystal (e.g., alkali halides, germanium, etc.) the polarization is always parallel to the electric field; then e reduces to a scalar. For a detailed discussion of the influence of macroscopic symmetries on the number of independent components of e and i, refer to BIRMAN'S article in Vol. XXV /2b of this Encyclopedia.

b) Microscopic aspects. The interaction energy of the electric field, E, and the crystal in the semi-classical and homogeneous approximation (i.e., without spatial dispersion) has the form

(2.3)

where M is the dipole moment of the crystal (Sect. 7). Both the ions and the electrons contribute to M.

For simplicity, it is useful to give a shorthand diagrammatic description of the main processes (Table 2.1). The three types of excitations: photons, electronhole pairs, and phonons, are represented by wavy, dashed, and solid lines, respectively. The interaction with light is characterized by linear dipole moments Ml (electrons) and Ml (phonons) as in diagrams d and e of Table 2.1.

The diagrams in the tables of this section will be identified with certain expressions for propagators (Green functions) and vertices in the framework of many-body theory. A detailed discussion of phonon Green functions is given in Sect. 34. For our present purposes, it is sufficient to know that every line represents a certain quasi-particle state with a well-defined energy and a sufficiently long lifetime or smalllinewidth. The number of vertices corresponds to the order in perturbation theory. The interaction potentials, V,:, are classified by two numbers, an upper index n denoting the number of participating


Table 2.1. Graphical representation of propagators of elementary excitations and linear couplings between them. Lower index: number of phonon lines, upper index: number of electron lines

~ Photon with frequency a, wave vector q, a a- (q) and odd parity ( - I)

Phonon with frequency w, wave vector q, b

wt (q) and even ( + ) or odd ( - ) parity (if present)

k E

Electron-hole excitation with energy e, =---c =-- and wave vectors k, k' k' e(k, k')

MI Linear photon-phonon interaction, d ~ Phonon dipole moment MI a- w"

MI Linear photon-exciton interaction,

e ~----a- e -

Excitonic dipole moment MI

Table 2.2. Graphical representation of some low-order anharmonic (cubic and quartic) phonon processes

Q First-order quartic renormalization a of phonon frequencies

w V4 w

b 2v3 Second-order cubic renormalization of phonon frequencies

w V3 w

W2

~ Second-order cubic contribution c

to phonon self-energy w w

WI

W3

d WV4~V4 Second-order quartic contribution w to phonon self-energy

WI

VT

e T 3 Cubic

thermal first -order frequency shift VT

f 1\ Quartic

Sect. 2 Outline of the theory of infrared absorption and Raman scattering 7

electron-hole pairs and a lower index m representing the number of phonons involved. The sum n + m gives the total number of lines joining at a vertex V':;.

In Table 2.2 some of the lowest-order processes of anharmonic interactions between phonons are shown (V,,? == Vm ; m = 3,4). Diagrams a, b, c, and d describe the first- and second-order contributions to the phonon self-energy, while the last two diagrams characterize the shift of the phonon energy due to thermal expansion or external fields. All diagrams are important to define the pseudo-harmonic approximation of lattice vibrations (Sect. 3). In addition, diagrams c and d depict lifetime effects which are essential to our discussion of infrared and Raman spectra in Sects. 8 and 9. For a complete discussion of phonon self-energies and interactions see Sects. 36 and 37.

If one replaces one or more phonon lines in diagrams c and d by the line for an electron-hole pair, one is led to non-linear electron-phonon processes. The significance of these processes follows from Table 2.3. Here, the two-phonon infrared absorption processes are shown in comparison with the corresponding one-phonon Raman scattering processes, where one of the final phonon states in the absorption process has been replaced by a scattered photon. The three different types of processes differ from one another by the number of phonons taking part in the non-linear scattering processes determined by V;_n (n=O: three phonons, n = 1: two phonons, n = 2: one phonon). Diagram 2.3 a describes the typical infrared absorption via an infrared-active Reststrahlen or dispersion

Table 2.3. Relation between first-order Raman scattering and second-order infrared absorption

Absorption processes Prototypes I phonon Raman Prototypes scattering

wi Two-phonon infra- wt One-phonon ionic red absorption Raman effect via

~ in alkali halides ¥-£ Reststrahlen -

a via Reststrahlen-IX Ml f1 wi"

oscillator Ml f1 oscillator (aiR)

w~ Ml g'-

± Two-phonon infra- wt One-phonon

Wi red absorption in (mixed)

a -c homopolar crys-

a '-£ Raman effect via b 1\IV\()o~- tals (Ge, Si, etc.) P \I\l\10-- non-linear dipole

Ml v1 via non-linear ~ wi" moments (M2 ) ----- dipole moments >::JM2 wi

(M2) >::JM2 Ml g'-

wi wt One-phonon e1ec-Phonon assisted tronic Raman

g- Blr electronic transi- g- Blr effect in homo-c f\/V\Cr- tions (vibronics) y~-- polar crystals

Ml Vi , ___ e. g. one-phonon Ml Vi~ (Ge etc.) and

ei sidebands of Ml g'- impurity-induced impurity centers effect in alkali

halides


oscillator with frequency W R, in ionic crystals. The process depicted in Diagram 2.3 b is the only possible absorption process in homopolar crystals such as silicon, where an electronic polarization decays into two (or more) phonons due to non-linear electron-phonon coupling. Usually this process is discussed in terms of the so-called non-linear dipole moment M 2' Diagram 2.3 c describes vibronic processes with an excited electron in the final state. These phononassisted electronic transitions are basically not part of our subject, but we include them here since the electronic Raman effect, as sketched in Diagram 2.3 y, is strongly related to these processes, as can be seen by inspection of Table 2.3. The other Raman scattering transitions described in Table 2.3 have not been observed since they contain at least one intermediate phonon, which strongly decreases the scattering efficiency.

The processes outlined in this section with the help of Tables 2.1, 2.2, and 2.3 will be discussed in detail in the following sections. After a summary of the properties of phonons in the pseudo-harmonic approximation (Sects. 3-6) we consider the theory of photon-phonon and photon-electron interactions first in the harmonic dipole approximation ("polaritons", Sect. 7), and then in the more general form needed for an analysis of infrared and Raman scattering processes (Sects. 8 and 9). The dynamic theory of lattices is presented in Chap. D in order to obtain explicit formulae for the infrared absorption (Sect. 12) and Raman scattering (Sect. 13) in crystals of certain symmetries and those as disturbed by external pressure or electric fields. A discussion of experimental spectra in various types of crystals using different degrees of approximation follows (Sects. 15-19). Emphasis is put on the model aspect of the theory which gives a semi-microscopic picture of the processes involved for different types of crystals, e.g., ionic or homopolar ones. Chapter F covers the phenomena observed in crystals which contain impurities. While some effects are similar to those in perfect crystals, many new types of processes are possible. The Lifshitz formalism is used in the description of perturbed lattice vibrations (Sect. 22) and Green functions are introduced (Sect. 23) to permit a condensed and straightforward discussion of localized and gap modes, resonances, impurity-induced spectra, etc. Concentration effects are treated in Sect. 28 and anharmonic processes in Sect. 30. As a natural extension, the absorption of mixed crystals (Sect. 29) as well as that of finite crystals (Sect. 33) is briefly reviewed.

The last chapter, G, deals with the formal aspects of phonons in terms of phonon propagators. Section 38 contains a collection of important formulae which are used in the discussions in previous sections.

The symmetry properties and selection rules of one-, two-, and threephonon processes in crystals of high symmetry are summarized in Sect. 39. The article ends with an alphabetic list of references.

Summarizing, we would like to state that this article does not intend to give a complete account of infrared and Raman processes in crystals. Instead, we hope to present a guide to the important features in the field and to provide useful tools for the interested reader.

Sect. 3 General properties of phonons 9

B. Phonons in insulators

3. General properties of phonons. Lattice vibrations or their quantummechanical analogue, phonons, are excitations in a solid which are due to correlations between the displacements of the ions from their equilibrium positions. Microscopically, they originate from the Coulomb forces between electrons and nuclei but with a strong modification by the quantum-mechanical uncertainty law and the Pauli principle for the electrons. This leads to rather complicated effective forces between the ions, and many approximations (e.g., the adiabatic approximation and/or models, e.g., the rigid ion model) are usually introduced in order to simplify actual calculations of dispersion curves and related properties. The heuristic distinction between approximations and models is useful in the sense that an approximation treats the complete Hamiltonian with an approximate wave-function (in order to obtain all possible effects qualitatively correctly) while a model starts from a simplified Hamiltonian and looks for rigorous solutions (to obtain some of the essential features quantitatively correctly). Usually, in 'both cases microscopic or macroscopic parameters have to be used which are fitted to experimental data, and very few a-priori calculations exist where a satisfactory parameter-free description of lattice vibrations has been obtained (refer to Sect. 6a). Thus, the determination of dispersion curves generally relies upon some experimental information. The most direct and, therefore, the most powerful method is inelastic neutron scattering which gives phonon frequencies of all wavelengths with an accuracy of a few percent. The method requires appropriate and rather expensive facilities and sometimes fails for high-frequency modes or for strongly incoherent scattering materials.

Infrared and Raman spectra allow the determination of many long-wavelength optic modes in crystals with an accuracy of better than 0.1 %. Until the beginning of the last decade this was, in combination with measurements of the elastic and optic constants, almost the only way of obtaining reliable information about the lattice vibrations of crystals. The phonons with shorter wavelengths appear in the second-order part of the spectra which usually exhibits continuous bands and can only be analyzed in a few simple cases in terms of combinations of single phonons. Therefore, in general, one needs additional information from other experimental sources, in particular from inelastic neutron scattering.

From the neutron or optical scattering data of a crystal a set of force constants can be determined, fitted more or less to the experimental data. With the force constants, the basis for calculations of one-, two-, or multi-phonon densities is established. In principle, the higher accuracy of the optical data as compared to those of neutron measurements makes it possible to check and refine the original set of force constants. However, anharmonic frequency shifts and broadenings in the phonon bands lead to difficult iterative procedures in determining the force constants. At present, such procedures have unexplored or unsolved theoretical and practical limitations.

10 Phonons in insulators Sect. 3

It is, therefore, of great importance for the analysis of phonon spectra to start with a well-grounded theoretical treatment of the lattice vibrations in the harmonic approximation. We shall give a condensed survey of the present state of the theory of phonons in insulators in this chapter, with emphasis on some important aspects of infrared and Raman spectra.

A discussion of the general properties of phonons follows, in particular, of the basic requirements for the existence of phonons, i.e., the dynamic stability of a solid (Sect. 3). We then review the different models which are used for ionic crystals ("shell" models, Sect. 4) and those for strongly covalent solids (valence forces, bond charges, etc., Sect. 5). Following this, the microscopic theory of phonons is outlined (overlap theory, dielectric-function method, Sect. 6). The chapter ends with a discussion of bonds, effective charges and effective fields providing a basis for the subsequent chapter on the interaction of phonons with external fields.

a) Dynamic and thermodynamic stability of solids. Classical electrodynamics cannot explain the fact that ions in a solid keep large mean distances of the order of Angstroms, while the attractive Coulomb forces between electrons and nuclei suggest a collapse of the whole system into a very dense cluster with particle distances of nuclear dimensions. Quantum mechanics tells us that this phenomenon is not caused by a breakdown of Coulomb's law at very small distances but follows from two non-classical principles:

1) The uncertainty principle which implies that the kinetic energy of electrons is enhanced if their charge distribution is compressed;

2) the Pauli principle which prevents two or more Fermi particles from simultaneously going into the same quantum-mechanical state.

The first fact is qualitatively known from the familiar description of a hydrogen atom in its ground state where the electronic charge distribution is essentially confined to a spherical shell with Bohr's radius as a mean halfdiameter.

The second point is the fact that without the "Fermi pressure" of the electrons, matter would not be stable. In other words, a material system in which electrons are replaced by boson-like particles would collapse into a very high density cluster.

While these two arguments are intuitively appealing, a detailed discussion requires some rather subtle arguments. We refer the interested reader to a recent review by LIEB (1976) and add only a few comments which are important for our further discussions. We mention the problem of the thermodynamic stability of solids which, in addition to the uncertainty and Pauli principles, requires the screening of long-range forces due to charge neutrality.

ex) Thomas-Fermi theory. An interesting insight into the problem arises from applying the statistical theory given by THOMAS (1927) and FERMI (1927). It was shown by TELLER (1962) that molecules do not bind in this theory and, furthermore, that the theory is asymptotically correct for Z --+ 00 (LIEB and SIMON, 1977). The idea of the Thomas-Fermi theory is to replace the unknown ground state energy of the system by a density functional. Let us suppose that


the system is built up from N electrons with k static nuclei of charges Zj>O and locations Rj corresponding to the Hamiltonian

N N

HN= L [-L!i-V(ri)]+ L Iri-r)-l+U({Zj,R)), (3.1) i= 1 i<j

with k

V(r;) = L Z)ri-R)-l, j= 1

U({Zj,R)) = L ZiZ)Ri-Rjl-l. i<j

The kinetic energy of the nuclei is not considered. Now, in the Thomas-Fermi theory a functional is defined (LENZ, 1932)

E(p)=2- 2/3 KeS p(r)5/3d3r

+~ H p(r) p(r') Ir _r'I- 1 d3 r d3r' (3.2)

- S V(r) p(r) d3r+ U({Zj' R)),

with its four terms representing the kinetic energy of the electrons, the electronelectron and the electron-ion interaction, and the nuclear-nuclear Coulomb energy. While the last two terms are exact, the first two terms are statistical approximations. The real ground state energy of the system E~ is then approximated by a minimizing (non-negative) function p(r) which is normalized with respect to a fixed number of electrons A leading to a Thomas-Fermi ground state energy

(3.3)

The minimizing density function, oy (r), is uniquely determined with A ~ Z k

=LZj' S p .. (r)d3r=A, and the equation for the chemical potential -p,~O 1

(3.4)

where p, is a convex, decreasing function of A which varies continuously from + ctJ to 0 as A varies from 0 to Z. Furthermore,

5j3Kc 2-2/3 pY(r)2/3cx:Zilr-Ril- 1 near Ri

and in the neutral case (A = Z)

(3.5)

r6 pTF (r) ~(3jn)3 [i Kc 2 -2/3]3 = const. (3.6)

as r~ 00. Equation (3.5) does not depend on the nuclear configuration {RJ which means that at large distances r, the information about the nuclear configuration disappears.

It can be said that the Thomas-Fermi theory is a theory of heavy atoms (i.e., large Z). The cores shrink with Z1/3 as N ~ 00 while in the outer part the mantle has a density pTF cx:r- 6 independent of Z. The third region is a transition region between the core and the outer shell where only a small fraction of the total number of electrons may be found while the outer shell is the


region in which binding takes place and which is followed by a last region in which the density exponentially vanishes. Only the two inner regions are relevant to the Thomas-Fermi theory. The above-mentioned "no binding" effect may be expressed by stating that in the Thomas-Fermi theory a big molecule is unstable under every decomposition into smaller molecules. The nuclear repulsion (U) is essential for this result since the inequality implied by the foregoing sentence is reversed if U is excluded from the Thomas-Fermi energy. The importance of the above results of the Thomas-Fermi theory lies in the fact that E~F, with suitably modified constants, gives a lower bound to the true quantum energy E~ for all Z. As a consequence, E~ is bound by an extensive quantity, the total number, N, of electrons, which does not depend on the nuclear configuration {R j }. This so-called" H stability" of matter has to be distinguished from the thermodynamic stability (see below). Including nuclear kinetic energy raises the total energy and does not change the result.

The stability of bulk matter can be expressed (DYSON and LENARD, 1967; LENARD and DYSON, 1968; LIEB, 1976) by the following lower bound for the ground state energy of N electrons and k nuclei with core charges Zi

( 1 k ]1/2)2 E~> -(const)N 1+ [N i~lZ~J3 (3.7)

> -(const) 2· (N +k),

where N + k is the total number of particles. Since the lower bound is at most proportional to the total number of

particles there exist "saturation" and stability of bulk matter. For bosons the result is (DYSON and LENARD, 1967)

-(const) N 1 /3 ~E~(bosons)~ -(const) N 5/ 3• (3.8)

This means that the ground state energy per particle, EUN, has no constant lower bound but grows proportional to N 1 ± 1/3. Bosons are, therefore, not stable under the action of Coulomb forces.

As a result, it turns out that the stability of matter requires not only the uncertainty principle for one particle (in the rigorous form as given by SOBOLEV (1938)) but, in addition, Fermi statistics, which ensures that the kinetic energy increases as the 5/3 power of the fermion charge density.

f3) Thermodynamic stability. So far we have discussed systems with N fermion particles keeping the total number finite. We would like to obtain a well-defined limit for EUN with N -+ 00 in order to give a precise meaning to all thermodynamic functions, such as temperature, pressure, and the free energy per particle for non-vanishing temperature. The introduction of this thermodynamic limit requires a "taming" of the long-range Coulomb potential ocr-1. A potential which decreases ocr -n with n < 3 + I: does not lead to a thermodynamic limit. The only possibility in the case of Coulomb forces is the cancellation of repulsive and attractive forces due to charge neutrality which should lead to a fast vanishing of the resulting forces at large distances.

Let us define the system by a Hamiltonian H N which contains the kinetic energies and interactions of n electrons and k nuclei of charge Z e, so that


charge neutrality means n=kZ. All particles are confined to a domain Q with volume V. The basic thermodynamic quantity is the partition function which, for convenience, may be chosen to be that of a canonical ensemble (defined by N =n+k V, and T)

Z(N, T, V)=Trexp(-PHN), P=1/kT, (3.9)

with the free energy per unit volume

F(N, T, V)= _(PV)-1 lnZ(N, V, T). (3.10)

The existence of F means the existence of a limit

~im Fj=~im F(Nj, Qj' p)=F(p, P), (3.11) )-+00 )--+00

where the integer index j denotes a sequence of systems with numbers of particles Nj=nj+kj and corresponding domains Qj (for example, balls with radii R). The same thermodynamic limit holds for any sequence Nj, Qj' if the thermodynamic limit of the density exists

lim Pj=lim N/Qj=p, j--+oo j--+DO

(3.12)

The proof (LIEB and LEBOWITZ, 1972) is carried out in two steps. The first shows that Fj has a lower bound. The second demonstrates that the sequence Fj is decreasing as j -+ 00, leading to the thermodynamic limit. Two essential ingredients of the proof are:

1) the Peierls-Bogoljubov inequality

TreA+B~TreA exp{Tr B eAjTreA}

and, most importantly,

(3.13)

2) Newton's integral for the Coulomb potential density, with p(r)=O for r>R,

of an isotropic charge

<1>(r) = J p(r') Ir - r'l- 1 d3 r' =r- 1 J p(r') d3 r' for r>R. (3.14)

Thus, <1>(r) = 0, if the charge distribution is neutral. The thermodynamic limit is obtained here for real matter where all the

particles are mobile. The proof holds also for "jellium", i.e., a material system in which the positive nuclei (or rigid ions) are approximated by a fixed, uniformly distributed background of positive charge (LIEB and NARNHOFER, 1975). A more realistic model for a solid is that in which the fixed nuclei are represented by point charges. Here, a final proof has not been given (LIEB, 1976) since local rotational invariance is lost which is required for the validity of Newton's integral, (3.14). There is, on the other hand, little doubt that such a proof must exist at least for a wide class of solid materials. First of all, crystals are built up from neutral cells which can only exhibit dipole moments, or, if the local point group symmetry is Td or higher, quadrupolar or higherorder multi poles. In the latter case the interaction potentials between subdomains fall off faster than r- 3 at equilibrium, but dipole moments induced by ionic displacements have to be considered. Now, for many crystal structures the Coulomb interactions have been summed up in terms of Madelung sums


and, in addition, stability conditions for compressibility, and shear moduli have been derived (see BORN and HUANG, 1954). These calculations are performed at a fixed lattice constant (i.e., density Po), and temperature treating the ionic displacements as small perturbations of the lattice energy. The existence of a free energy F(p, T) in the thermodynamic limit is, at least, implied by all quasi-harmonic and self-consistent phonon theories. If the temperature is, however, very close to that of a phase transition the critical behavior of the crystal requires a more subtle treatment of its dynamic and thermodynamic properties. While this topic is generally outside the scope of this article, we shall consider it to some extent in the case of the ferroelectric soft modes in ionic crystals. Here they are important for the understanding of the interaction of phonons with external fields.

Before leaving this field we note a few more general properties of stable matter in the thermodynamic limit which are important for the following:

1) Definitions (3.9) and (3.1 0) show that for every finite system In Z (N, V, T) is convex in fJ=(kT)-l and hence fJF(N, V, fJ) is concave. Therefore, the limit fJF(p, fJ) has to be concave. Correspondingly, the specific heat - fJ2 02 fJF(p, fJ)/oP ~ o.

2) Thermodynamic stability in a strict sense is defined by the requirement that F(p, fJ) be a convex function of p. It means that the compressibility

(3.15)

Therefore, in contradiction to approximate theories such as the van der Waals theory of the vapor-liquid transition or other model theories where polarization or magnetization plays the role of p, F should not show a double bump. Therefore, the ground state energy E(p) is also a convex function of p.

Since F is convex in p and concave in fJ in a (p, fJ)-plane it is jointly continuous in (p, fJ) which implies that the thermodynamic limit is uniform on (p, fJ) sets which are bounded above and below. For E(p), the same result holds.

One has to note that instead of the canonical ensemble {N, V, T} used here, the micro canonical ensemble {N, V, E} and the grand canonical ensemble {Il, V, T} both lead to the same values for thermodynamic quantities.

The existence of a thermodynamic limit does not mean that a single phase system with uniform density has to exist; e.g. a solid phase (with Ps>p) may co-exist with a gas phase (with Pg < p). Therefore, the thermodynamic limit does not establish a unique thermodynamic state if several phases exist. This means that correlation functions defined for a finite system do not necessarily have unique limits as V ~ 00. In spite of these general difficulties we shall assume in the following chapters that the solids under discussion are in a well-defined thermodynamic state so that all correlation functions are uniquely defined with respect to the Hamiltonian H and the equilibrium free energy F(p, fJ) = F(V, T).

If we accept this as the basis for the calculation of the motion of the ions and electrons from the many-body Schrodinger equation we still face a practically insoluble problem. Important simplifications can be made for the following reasons:

IX) the masses of the ions are <: 104 larger than those of the electrons. This leads to the adiabatic approximation (refer to Sect. 3b) in which the lattice


forces are treated as effective ion-ion forces while electronic coordinates are eliminated;

p) the displacements of the ions (except for quantum crystals) can be treated in the harmonic approximation (Sect. 3.c) at sufficiently low temperature. This allows the use of classical mechanics for the evaluation of the eigenfrequencies of the crystal;

y) the range of the effective non-coulombic forces is restricted to near-neighbors and few partners, and, two- and three-body forces are usually sufficient. Coulomb forces need a specific summation technique which takes advantage of charge neutrality. Therefore, convenient boundary conditions ("infinite" crystal or periodic boundary conditions) may be applied which allow the full use of symmetry properties, in particular crystal periodicity, which reduces the problem finally to that of a classical few-particle system similar to a molecule.

In the following sections we shall elaborate upon these points in the course of our discussion of properties of lattice vibrations.

b) The adiabatic approximation (BORN and OPPENHEIMER, 1927; BORN and HUANG, 1954). The big mass difference between the nuclei and electrons causes the corresponding clear separation between the low-energy lattice vibrations and (the majority of) the high-energy electronic excitations in which the nuclei are practically at rest. This separation enables us to concentrate on the energy regime of lattice vibrations and to assume that real electronic excitations may be neglected in the following.

The forces connected with lattice vibrations are usually considered as if they acted between the ion cores. This formal treatment of the lattice forces as effective ion-ion forces is based on the adiabatic approximation, in which the electrons are assumed to follow any displacement of the cores instantaneously by taking up a force-free equilibrium configuration. Thus, in the complete set of coupled equations of motion for cores and electrons, those equations describing the electrons reduce to subsidiary conditions which allow complete elimination of the electron coordinates. This concept is tacitly assumed in all formal treatments, and it is explicitly used in the dipole models such as the shell model (see for example, COCHRAN and COWLEY, 1967; ZYBELL, 1972). It seems that the adiabatic treatment is valid in large-gap insulators because the energy of phonons is of the order of 10- 3 of the gap energy, i.e., of the minimum electronic excitation energy from the ground state. For metals it has been argued (CHESTER, 1961) that Pauli's exclusion principle prevents most of the conduction electrons from taking part in low-energy transitions which could contribute to a non-adiabatic electron-phonon coupling. This argument has been given a more quantitative form by BROVMAN and KAGAN (1967, 1975). These authors showed that the non-adiabatic corrections to the phonon frequencies in metals are at most of the order of the energy of the Debye frequency divided by the Fermi energy, i.e., ~10-3. Apparently, the Fermi energy in metals acts as an effective electronic energy gap for the majofity of the phonons. It is only in a small part of the Brillouin zone and under special circumstances that the corrections may be larger than 10- 3. SHERRINGTON (1971) has shown that even in a-Sn, which is an intermediate case between real


insulators and metals, non-adiabatic effects are negligible, aside from a small energy regime at long acoustic wavelengths (see also GIULJ and PICK, 1974).

While the adiabatic approximation for the phonons in its principal aspects (i.e., the large mass ratio between electrons and ions) is now a part of standard textbooks, it requires subtle argumentation when analyzing the electron-phonon coupling, for example in Raman scattering. We shall therefore outline some essential formal details of the adiabatic approximation in Sect. 9 where they will be needed for the discussion of Raman scattering.

c) Force constants (LEIBFRIED, 1954; MARADUDIN et aI., 1971; MARA

DUDIN, 1974). The lattice vibrations of most crystals can be described at low temperatures by assuming a lattice potential in the adiabatic and harmonic approximation, i.e., a potential which depends on small displacements of the ions in a bilinear form only. Also, the expansion coefficients are supposed to contain the anharmonic effects of thermal expansion of the lattice and phonon self-energy by a renormalization procedure. This leads to the definition of a set of pseudo-harmonic force constants at a given temperature T and volume V which are continuous and usually monotonically decreasing functions of T and V (refer to Sect. 3e). We shall now describe the main properties of lattice vibrations within this approximation. The classical, harmonic lattice potential has the form (cf. (34.5))

¢2=1II I Ua(IK) ¢ap(IKll'K') Up(l'K') (3.16) ap Ii' ",,'

=1 I u(L) ¢(LI!,) u(I!,) L,L'

=1u¢u.

(3.17)

(3.18)

Here, a and f3 are cartesian indices, L is short for the cell index 1 and particle index K of the n different ions in a cell. The respective equilibrium positions of the ions in the cell I are defined by the lattice vectors

x(L) = X(IK) = x(l) + X(K). (3.19)

x(l) is pointing to the I-th cell of the crystal and X(K) gives the relative position of the K-th ion in the cell. The general force constant matrix ¢ consists formally of two-body force constant matrices ¢(L, L) and is subject to the conservation laws of energy, momentum, and angular momentum which are a consequence of the symmetry restrictions of the space group (see Sect. 3e and, for details, BIRMAN'S article in Vol. XXV /2 b). The bilinear form of the potential ensures that we can treat the equations of motion classically in order to obtain the correct energies of the system (see, for example, MESSIAH, 1964).

The vector equation of motion for an ion at site x(L) with mass M" is

MLii(L) = - a~tL) = - t ¢(L, I!,) u(I!,), (3.20)

(ML=M", independent of I),

which can be solved using periodic boundary conditions for a crystal with N cells. With the help of the conservation laws and lattice periodicity the 3 nN

Sect. 3 General properties of phonons

equations reduce to a set of 3iz homogeneous equations

w2(qj) e(K I qj) = L: DO(K K'I q) e(K'1 qj) ,,'

by a Fourier transform for the displacements u

17

(3.21)

u(L) = (NM L)-1/2 L: Q(qj) e(KI qj) exp{i(q x(L)-w(qj)tn (3.22) qj

and, similarly, for the force constant matrix cp

DO(KK'lq)=(M"M",)-1/2 L: cp(L, E) exp{-iq[x(L)-x(E)]}. (3.23) I

Here w 2 (q j) == w~ are the eigenvalues of the dynamical matrix DO for a given wave vector q. The branch index j labels the 3n solutions which (accidentally or by symmetry) may be two- or threefold degenerate. The eigenvectors e(KK'lqj)==B(KK'IA), for fixed q, form a complete, orthogonal set with 3n components. Q(qj)==Q(A) are the 3nN normal coordinates which are the basis for the introduction of phonon creation and annihilation operators in Sect. 10.

Whereas (3.21)-(3.23) give the most convenient representation for the general symmetry-related properties of phonons, it is, for the purpose of the model theories discussed in forthcoming sections, sometimes advantageous to use a slightly different formulation. In the shell model and its extensions, the adiabatic behavior of the electronic motion requires an explicit expression for the particle masses. We shall then use, instead of (3.22) and (3.23) the following Fourier transforms:

u(L)=N- 1/2 L Q(A) U(KIA) exp {i(q X(L)-WAt)} A

and

D(KK'I q)= L cp(L, E) exp {-iqx(L, En I

= (M" M"l/2 DO(KK'I q)

which leads to

M"w~ U(KIA)= L D(KK'lq) U(KIA) ,,'

or, in a short-hand notation,

W~MU(A)=D(q) U(A).

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

Here, D is a dynamical matrix built up from the n2 submatrices D(KK'I q) with 3 x 3 cartesian elements D"p(KK'1 q). M = (M,,) is the diagonal mass matrix. The hypervectors U(A) consist of the n vectors U(KIA)=(M,,)-1/2 e(KIA).

In an actual case one often wants to determine a set of force constants from a restricted number of eigenfrequencies w A' usually measured by neutron spectroscopy in a limited number of directions in q-space. With the help of this set, one can determine the complete spectrum of frequencies W A• The onephonon density of states, P(w)=L(j(w-w..), as well as multi-phonon densities

A

can be calculated, too. These densities are essential for the computation of the


specific heat of the crystal and the infrared and Raman cross-sections of light. For the latter a complete knowledge of the eigenvectors e(KIJe) is required. Unfortunately, the eigenfrequencies OJ A determine the direction and relative length of the eigenvectors e only for certain symmetry points in the B.Z. In general, as discussed in detail by SZIGETI and co-workers (LEIGH et aI., 1971), (3.21) can be transformed continuously into an equivalent equation of the form

OJi • e' (A) = D' (A) e' (Je) (3.29)

by a unitary transformation T which gives D' = T+ DT and e' = T+ e with unchanged eigenfrequencies OJ A as long as T fulfills certain symmetry restrictions. The argument is valid also for the model treatment discussed in the following sections, and it shows that a set of (formal or model) force constants derived from experimental data has to be treated with caution because its physical significance might be doubtful. Additional arguments are necessary to prove that interaction parameters fitted to experimental data are more than a lucky simulation of the real mechanism. This is one of the main difficulties of model theories (Sects. 4, 5), in particular in connection with Raman scattering (Sect. 18). A measurement of eigenvectors in low-symmetry directions, which is possible in principle, would be helpful in this situation.

d) Symmetry properties of phonons. The analysis of dynamical properties of crystals such as infrared and Raman spectra employs group-theoretic methods in order to obtain selection rules, etc. A comprehensive discussion has been given recently by BIRMAN in Vol. XXV/2b of this Encyclopedia (1974a). Useful short accounts may be found in a recent book by LAX (1974) and in a condensed article by BIRMAN (1974b). There is no need for a repetition of these elaborate treatments. Some useful tables of symmetries and selection rules in one-, twoand three-phonon processes in crystals with high symmetry are presented in Sect. 40. We shall use, with some slight and obvious modifications, the notation given by BIRMAN (1974a, b).

IX) Critical points. An important subject in the analysis of phonon spectra are the critical points which often appear as characteristic features in the measured bands and sometimes are helpful in identifying specific phonons or phonon combinations by using suitable experimental techniques such as differential Raman scattering.

The critical points occur at specific wave vectors, qc' where the energy surfaces (as defined by the values of OJ2(q j) of a branch j) have vanishing slope. The normalized fraction of squared frequencies in the interval (OJ 2, OJ2 + dOJ 2)

for the branch j is

( 2 d 2 SS dSq 2 Nj OJ) OJ =Vo IVq OJ 2 (qj)ldOJ (3.30)

where dSq is a surface element of a constant OJ-surface in q-space. For the interval (OJ, OJ + dOJ) we can obtain a similar formula for the branch density niOJ) by the equality

(3.31)

Sect. 3 General properties of phonons

Anomalous behavior of the density of squared frequencies

N (w2) = I ~(W2), j

may be expected at wave vectors qc for which

VqW2(qj)q=qc =0

19

(3.32)

(analytical critical points), or changes sign discontinuously (non-analytical critical points). The theory of these critical points (VAN HOVE, 1953; PHILLIPS, 1956; ROSENSTOCK, 1955, 1957) leads to the following general results:

1) For analytical critical points P; with index i in an s-dimensional q-space a Taylor expansion exists in a region around qc'

s

W2(qj)=W2(qcj)+ L Brij} . (3.34) r= 1

with By = ± 1, where i is the number of negative Br In this expansion and qrocl(q-qJI is defined in a locally transformed system.

2) In an s-dimensional Brillouin zone there exist, for each branch j, at least

G) analytical critical points, p;. These different P;'s are related by topological

conditions (Morse relations) which determine a topological set of critical points. 3) The cubic or lower symmetry of the space-time group G(q) leads to a

definite number of critical points which corresponds to a symmetry set. 4) The actual minimal set of critical points consists of the smallest topologi

cal set which contains the symmetry set. These are in one, two, and three dimensions for a single branch j:

Il() I-d: one Po (minimum), one ~ (maximum); [3) 2-d: one Po (minimum), one Pz (maximum),

two Po (saddle points); y) 3-d: one Po (minimum), one ~ (maximum),

three ~ (saddle points), three Pz (saddle points). 5) The analytical behavior of the density of squared frequencies Niw2) in . . (( {I for x;;::: 0) one, two or three dimenSIons IS e x)= 0 for x~O

.J 1-d: At ~J: N(w2)oce[±(w2-wDJ IW2_W~I-1/2 [3) 2-d: At ;: N(w2)oce[±(w2-w~)J

2

(3.35)

~: N(w2)oclnlw2-w~1

y) 3-d: At;' i}: N(w2)oce[±(w2-wDJ IW2_W~I+1/2. l' 3

For n(w), the equivalent equations are obtained from (3.31). 6) Non-analytic critical points cannot be described in terms of a regular

expansion series, (3.34), but their topological criteria are still the same as those of analytical points. For the analysis of phonon spectra, saddle points. are most


important due to their 'kink' or 'near-logarithmic singularity' peak behavior. The second case seems to be approximately realized in spectra of crystals with the sodium-chloride structure (refer to Sect. 15).

In differential spectra, where derivatives of the frequency densities appear, the structure of critical points is generally more pronounced. The characteristics of the 'differential' critical points follow easily from paragraph 5 since the calculation of dN(w2)/dw2 or dn(w)/dw requires only three functions:

(3.36)

and d In x/dx = 1/x.

f3) Selection rules. We consider the two optical processes which are most interesting in the context of this investigation, namely infrared and Raman processes.

In an infrared (dipole allowed) process the perturbation Hamiltonian is proportional to the induced dipole moment M of the crystal which depends on the wave vector of the light, k, and the perturbed configuration of the ions, R:M=M(R, k). This polar vector transforms (k-+O) as the representation of T1S in cubic crystals with inversion symmetry,

(3.37)

which has to be decomposed in, at most, three components in crystals with low symmetry.

For Raman scattering the induce~ charge deformation corresponds approximately to a symmetric second-rank tensor with 6 components which (for vanishing wave vectors of the incident and scattered light) in a cubic crystal with inversion symmetry can be represented by the direct sum of three evenparity representations (c.f. BIRMAN, 1974, p.244)

[D(r)J(2) = I;.+ EBI;.i EBr;~. (3.38)

If we have determined the symmetry of the initial state Ii) and of the final state If) we can determine the non-vanishing matrix elements <fl I;.s Ii) and <fl r,,+ Ii) with a projection operator technique. In the case of multi-phonon processes, it is first necessary to deduce the products of the representations of the different phonons involved in the transition. For further details we refer the reader to BIRMAN (1974a, b) and LAX (1974).

e) The pseudo-harmonic approximation. An analysis of experimental phonon spectra is usually made at a given temperature and volume (corresponding to atmospheric pressure). Since the equilibrium positions of the ions in crystals change with temperature due to anharmonicity, the crystal parameters, such as force constants and anharmonic coefficients, depend on temperature. If the complete crystal potential were known, we could obtain from this potential the thermal expansion of the crystal as well as the coupling parameters for every given temperature. This can be done to some extent for the rare-gas crystals, where a modified Lennard-Jones potential gives rather satisfactory results (see, for example, GL YDE and KLEIN, 1972). Usually, the crystal potential is un-

Sect. 4 Ionic crystals 21

known, so that it is convenient to split the anharmonic effects into two parts. The first part is related essentially to the thermal expansion of the crystal. In parameter-free crystals specified by a single lattice constant, such as the alkalihalides, this results simply in the change of the crystal volume (in more complex crystals there are sublattice displacements, in addition). We can then, for a particular temperature, begin with the Helmholtz free energy F(T, V) in order to obtain the crystal parameters belonging to these values of T and V (Sect. 38). We call these quantities obtained from a series expansion of F(T, V), quasi-harmonic parameters (LEIBFRIED and LUDWIG, 1961; LUDWIG, 1967). While these contain the changes due to thermal expansion the additional anharmonic effects at given T and V have to be calculated separately. With respect to a single phonon, these anharmonic interactions may be approximated by a complex self-energy which adds a frequency-dependent correction to the quasi-harmonic frequency (see Sects. 12d, 36c and 38).

Such a treatment will be called a pseudo-harmonic approximation (COWLEY, 1963; BILZ, 1966). It retains the idea of distinct phonons, i.e., excitations which are considered to be well-defined (self-consistent) quasi-particles with sufficiently long lifetimes (GOTZE, 1967). In practice, it turns out that cubic and quartic renormalizations up to second order (as indicated in Table 2.2) are in many cases sufficient to obtain reasonable agreement with the properly analyzed (peaks of) measured frequencies. There exist, naturally, general limits to this concept which are well known from Hartree-Fock treatments of quasi-particles (see, for example, COWLEY, 1965), but they do not affect the practical use for the cases under consideration.

In the following we shall tacitly assume that the phonon frequencies used in the calculations are obtained from a pseudo-harmonic treatment so that the differences between calculated and measured frequencies may be neglected. In some important cases, for example for dispersion oscillators in ionic crystals, the problem of a consistent calculation of their frequencies is explicitly discussed (Sect. 12). A related case is the analysis of the temperature dependence of lifetimes of dispersion oscillators (Sect. 15). Here we need a careful distinction between quasi-harmonic and anharmonic self-energy contributions. A detailed account of phonon dynamics with anharmonic interactions is given in Chap. G (Sects. 34-38), while important results are discussed in the context of specific problems. In addition to the book by BORN and HUANG (1954) the interested reader is referred to recent articles by BARRON and KLEIN (1974, perturbation theory of anharmonic crystals), HORNER (1974, strongly anharmonic crystals) and GbTZE and MICHEL (1974, self-consistent phonons).

4. Ionic crystals

a) The rigid-ion model. The main part of the lattice energy of ionic crystals can be described by the two-body Coulomb interactions of point-like ionic charges as was discussed originally by BORN (1923; see also Tosi's review, 1964). In diatomic cubic crystals with ions of charges ZKe= ±Ze, the equilibrium electrostatic energy per lattice cell is given by a sum over two-ion contributions


(4.1)

This expression defines the Madelung constant IXM ; ro is the nearest-neighbor distance. (For a table of values of IXM refer to BORN and HUANG (1954, app.2)). At equilibrium, the attractive forces resulting from this potential have to be balanced by the repulsive forces of the electrons in the overlapping outer shells. To a first approximation, this can be conveniently expressed by assuming two-body central Born-Mayer potentials y"x·(r) = Axx' exp( - r / Px.J, which yield a lattice energy given by

. (Ze)2 1 , U(r)= -IXM --+-2 L V"".(r) (4.2)

r K,1('

with the equilibrium condition

(4.3)

Using the free ion charges Z ~ 1, this Born model of ionic crystals gives good results for the cohesive energy of alkali halides (less than 5 % error). The contribution of the repulsive part is of the order of 10 %. Additional contributions come from many-body repulsive forces (Sect. 6a) and from van der Waals interactions which usually (but not in rare-gas crystals) contribute only a few per cent of the total energy so that to a good approximation they may be neglected (TOSI, 1964).

Born's model was the starting point for the dynamical treatment of alkali halides by KELLERMANN (1940). Corresponding to the two parts of the lattice energy there are short-range and long-range (Coulomb) forces so that the force constant matrix </J may be divided as

(4.4)

Let us, for example, consider the case of a sodium chloride lattice in a nearest neighbor approximation, where K' =(100) is the index of the relative position, L - L, of a nearest neighbor of any single ion, located at K = (000). In the following we label the positive ions by L=(l, +) and the negative ones by L = (1, -). Then, the symmetry of the cubic lattice in the rocksalt structure reduces the matrix </JR(L - L) = </JR(1 001 +, -), to the form:

(4.5 a)

(4.5 b)

Here, Vo is the volume of the elementary Wigner-Seitz cell. The factor e2/2vo is removed from A and B for convenient comparison with the Coulomb forces.

Sect. 4 Ionic crystals 1,.5

The dimensionless quantities A and - B have values between 1 and 20 for the alkali halides while the corresponding values of the Coulomb force constants are AC = -8 and BC =4.

The Coulomb force matrix is usually calculated by a specific summation technique originally due to EWALD (1917a, b) and later on improved by BERTAUT (1952) and others and extended to more general cases (DE WETTE, 1959). The procedure results in a macroscopic field which lowers the original point symmetry of the crystals and leads to a splitting of degenerate modes. For example, the three optic modes in a cubic diatomic crystal split into one longitudinal and two transverse modes. (We note that we discuss the phonons in the static limit above the polariton regime, i.e., Iql ~w/c, or, in the limit c --t CX)). Therefore, the Coulomb matrix has a non-analytical part at q = 0 which requires some care in calculations for low-symmetry crystals. The Coulomb force constant matrix can be represented by

cjJC=ZCZ, where Z=(ZI() (4.6 a)

is a diagonal charge matrix with the Coulomb matrix, after Fourier transform, Eq. (3.25),

C(KK'lq)= 4ne2 [q~q -Q(KK'lq)+bKK, L Q(KK" 10) ZZK"] , (4.6b) Vo q K" K

where the components of Q are obtained with the help of Ewald's Thetatransformation (refer to BORN and HUANG, 1954; MARADUDIN et aI., 1971, note the different notation). We shall discuss the properties of the macroscopic field E and the Lyddane-Sachs-Teller splitting of longitudinal and transverse modes later (refer to Sect. 6d). We note, that in a diatomic cubic crystal

Q(K K' 10)=F, I = unit matrix, (4.7)

which leads to the Coulomb matrix of the transverse optic mode (E = 0)

4nZ2 e2

3 IsgnKsgnK', Vo

(4.8)

While the first term of (4.6b) defines the (longitudinal) macroscopic field which is related to the so-called ion plasma frequency

Z2 e2

JlW~dq=0)=4n--, Vo

the regular term, Eq. (4.8), gives the transverse frequency WTO

2 2 4n Z2 e2 JlWTo =JlWo--3 -

Vo

(4.9)

(4.10)

where W& originates from the short-range forces, and the longitudinal frequency

(4.11)

We see that the long-range dipolar displacement correlations, which manifest themselves in the ion-plasma frequency, lift the degeneracy of wTO with wLO '

24 Phonons in insulators

Na! 700 K -- BREATHING-SI£U MODEL ---SIMPLE SI£LL MODEL [ntl .•••••. RIGID-ION MODEL ~ tOl ..... . '" ·T 7

• ••• 0 L ".

7 .......... [OO~]

6

.... .L •••• ~ T

". '"

• T I oL I

I ···.1 .... ....----

.1" ..... //

"()-'" .... ···V

". '"

6

0 0 5

4

................. --'---''---L-...... ~_..J...J_ ........ _.L-_.L--=-" L-..L-..L-......... -L.:::-' ° t- [o.Q~-t 1,0] ... ~ [!,UJ

REDUCED WAVE VECTOR

Sect. 4

Fig.4.1. Dispersion curves of Nal. Experimental data: WOODS et al. (1963). The different models are discussed in the text

The model introduced here considers rigid point charges and is called the rigid-ion mode1. The simplest version of this model is a nearest neighbor approximation for the short-range forces (KELLERMANN, 1940). The equilibrium condition (4.3) leads to

(4.12)

if B is assumed to be derived from a central potential V(r) as in (4.5b). The only free parameter, A, can be fitted to the bulk modulus (equal to the reciprocal compressibility /3)

1 02 UI 1 e2 BT=-:;z =-3 (A+2B)-. (4.13)

r ur r=ro 2vo

The results of such a simple model for NaI are shown in Fig.4.1 and compared with neutron spectroscopy data. The agreement is rather good except for the longitudinal optic (LO) branch. The electronic polarizability is neglected in the rigid-ion model (the high-frequency dielectric constant here equals unity, €~ = 1). Thus, although 1'0 and the transverse optic (TO) frequency at rare quite accurately found in this model, the rigid-ion LO frequency turns out to be too high since experimentally 1'00 > 1 and, according to the Lyddane-SachsTeller relation (LYDDANE et a1., 1941),


(4.14)

The structure of the dynamical matrix of this "Kellermann" model is generally

DR1(q) = R(q) + ZC(q) Z. (4.15)

Here Rand C are built up from the Fourier transforms of the short-range matrix cpR and the Coulomb matrix cpc, respectively. Z are diagonal charge matrices analogous to the mass matrices M.

Summarizing, the following features of the rigid-ion model are noted:

1) The interaction is described by pair potentials, (often central), between the charged particles. Since the short-range forces are essentially related to exchange effects between overlapping electron clouds, an overlap correction of the model is necessary (Sect. 6 a). This leads to non-central three- and four-particle potentials. For example, the n.n. force constant B may contain a non-central part BNC'

not determined by (4.12), due to unspecified many-body forces. 2) Since retardation effects are neglected, the long-wavelength limit (q-+O)

for the optical branches still corresponds to phase velocities w/q 4" C i.e. w/q much smaller than c, the velocity of light (WTO/C> 102 cm -1 in NaI). The macroscopic field E as determined by div(E+4nP)=0 is connected only to the longitudinal part of the displacements. This shifts the longitudinal optic frequency WLO to frequencies higher than WTO. One can describe this splitting of the long-wavelength optic branches by considering the macroscopic field E (with symmetry Coo) which reduces the crystal symmetry (BIRMAN, Vol. XXV/2b; MARADUDlN, 1976).

3) The regime of strong phonon-photon coupling, where q;::;;,wTO/c, essentially determines the features of infrared absorption and will be discussed in later sections. We note that the width of the reflection band is given by

(4.16)

while the rigid-ion model gives Wio(Bo -1). The electronic polarization determmmg Boo screens the interaction of the macroscopic field E with the longitudinal mode at W=WLO and diminishes the rigid-ion value of Llw2 by a factor Boo.

4) Effective "rigid-ion models" are sometimes fitted successfully to dispersion curves of alkali halides and II-VI and III-V compounds (KUNC et aI., 1975b). These models simulate a non-negligible electronic polarization by using static charges Z much smaller than unity and by introducing rather strong second-nearest-neighbor force constants (NAMJOSHI et aI., 1971). While a certain decrease in Z from 1.0 to about 0.9 might be justified by overlap effects (Sect. 6a), the use of values around 0.5 or 0.6 destroys the consistency of such a model with a rigid-ion calculation of the cohesive energy and indicates only a lucky compensation of short- and long-range effects (COCHRAN, 1971; VERMA, 1971). However, for ionic crystals with three or more particles in an elementary cell, effective rigid ion models are, at the present time, in most cases the only feasible way to describe experimental dispersion curves.


b) Dipole models. The adiabatic approximation (Sect. 3b) allows the complete elimination of the electronic degrees of freedom from the lattice potential. A description of phonons by a set of formal force constants, i.e., in terms of effective ion-ion forces is therefore always possible. On the other hand, the physical meaning of these sets becomes doubtful as the number of parameters increases. Furthermore, the origin of some effects, such as the lowering of the longitudinal optic modes discussed for alkali halides, is well known to be the electron-ion and electron-electron interaction. One might, therefore prefer a model description which first treats these interactions explicitly in a harmonic approximation and eliminates the electron coordinates via the adiabatic condition. This should lead to a reduction of the number of parameters needed, to a better understanding of their physical meaning, and to a clearer relation to the microscopic theory.

Following the early discussion by TOLPYGO (1948) and SZIGETI (1948, 1949), a number of models have been developed during the last 15 years which treat the displacement-induced distortions of the electronic charge density in a dipole approximation. A description of these models was given in the article by COCHRAN and COWLEY (1967) in this Encyclopedia and, with some recent developments, by CoCHRAN (1971), HARDY (1974), and BILZ et al. (1974). A short summary of the results is given for our purposes. We use here the most pictorial of these models, the so-called shell model (DICK and OVERHAUSER, 1958; COCHRAN, 1959).

We add to the displacement vector coordinate, u(L), of a single ion, specified by L=(IK), an electronic polarization coordinate, w(L). The potential depends now in a bilinear form on all the w(L) and u(L) corresponding to 2 x 3 nN degrees of freedom. The equations of motion are

Ml(ii(L) = - a~tL) (4.17)

mel w(L) = - a!fL) = 0 (adiabatic condition).

Introducing short-range and Coulomb interactions between electrons as well as between electrons and ions, and proceeding as in the two foregoing sections, we obtain from (4.17) the equations

w2 MU(q)=(R+ZCZ) U(q)+(T+ZCY) W(q)

O=(Tt+ YCZ) U(q)+(S+ YCY) W(q). (4.18)

The first two terms on the r.h.s. of the first equation correspond to the rigid-ion case, Sect. 4a. The matrices T and S represent the short-range electron-ion and electron-electron couplings. Y is an electronic charge matrix analogous to the ionic charge matrix Z. The shell model introduces a charged (~) but massless shell at each ion site, the displacement of which, relative to the ions, is the polarization coordinate w. By eliminating W from (4.18) we obtain

w 2 MU(q)=D(q) U(q) (4.19a)


with D(q) = DR!(q) + DDIP(q),

where DR! is the dynamical matrix of the rigid-ion model and DDIP that of the induced dipolar forces

DDIP= -(T +ZCY)(8+ YCy)-l (Tt + YCZ). (4.19b)

In the dipolar part DDIP of D, the matrix 8 consists of the short-range electronelectron coupling S and of the local shell-core coupling K" (i.e. a constant diagonal matrix K (for localized ionic polarizabilities)). Occasionally, intersite shell-core couplings are considered, too.

8(KK'1 q)=S(KK'1 q)+Kb"". + ...

K"p=b"p[K" + 4,,(K 1 O)-S",,(K 1 0)]. (4.20)

In the simplest version of the model one assumes that short-range forces act through the shells (R= T= Tt =S) only and furthermore neglects the polarizability of the positive ions by comparison to that of the negative ions (K+ ~ 00). The latter statement establishes a sufficient approximation for many alkali halides. In the nearest-neighbor approximation there are then two additional parameters, Y_ and K_, as compared to Kellermann's rigid-ion model. These parameters can be fitted to the optical constants in such a way as to give agreement for wTO and WLO by using the equations (refer to COCHRAN and COWLEY, 1967):

(4.21)

Here Jl. is the reduced mass of the lattice cell, and the effective Szigeti charge Zs (see Sect. 6d) is given by

(4.22)

and the "center" frequency, W; =H2wio+ eoo wio] , is

(4.23)

or (R~)-l =ROI +K=l, with Ro=A+2B.

This three-parameter shell model (A, Y_, K _) improves the agreement remarkably compared with the rigid-ion model, as can be seen in Fig. 4.1.

The Eqs. (4.21)-(4.23) suggest an alternative representation of (4.19) (COCHRAN,1971):

D=R' +Z'C' Z't, (4.24)


with the following effective quantities:

R' =R - TS- 1 Tt, an effective short-range matrix

Z' = Z - TS - 1 Y, an effective (Szigeti) charge matrix, and

C'=(I+CYS- 1 y)-lC, an effective Coulomb interaction,

where I is the unit matrix.

Sect. 4

(4.25)

(4.26)

(4.27)

The general structure of (4.27) looks very appealing, and it is completely analogous to the general structure of the microscopic theory (Sect. 6). The shell model Uust as the rigid-ion model) is, of course, able to describe every type of crystal dispersion relation by introducing a sufficient number of parameters in the matrices R, Sand T, but it is still deficient from a physical point of view. We shall see this by returning to the simple shell model for alkali halides. It is apparent from Fig. 4.1 that this model is in disagreement with the experimental data for certain regions of the B.Z., especially for the LO-branch in the neighborhood of the L point (2roq=n(1, 1, 1». Here, the shell model gives the same result as the corresponding rigid-ion model, since the polarizable negative ions and their shells are at rest. The corresponding motion is shown in Fig. 4.2.

The cause of the failure of the shell model lies in its omission of the distortion of the electronic charge distribution around the negative ions (see Sect. 4d). As has been shown (COWLEY et aI., 1963), this effect can be simulated by giving the positive ions pseudo-polarizabilities. They may be understood as the result of the polarization of the overlap charge (see Sects. 4e and 6a).

The shell model in this extended form has been widely used in recent years for many ionic diatomic crystals and also for a few multi-atomic ones (for details, see COCHRAN, 1971). Its main success is the correct description of the contribution of polarization-induced dipole forces. In this respect, the shell model is equivalent to the often used "deformation dipole model", developed by HARDY (1957; KARO and HARDY, 1980), and to TOLPYGO'S model (1948). Both models contain a dynamical screening of the static ionic charge Z to give an effective charge Z', see (4.26), and take into account the electronic polarizability,

(4.28)

in the shell model. Nonetheless, they do not contain a "deformability" of the ions implying that a genuine short-range deformation takes part in the in-

Fig. 4.2. LO Breathing mode at q=(1, 1, 1)n/2ro


teraction (corresponding to R - R' = TS - 1 T t, (4.26), in the shell model). This omission causes certain difficulties, e.g. in fitting the optic frequencies (OTO and (OLO' since they depend on this effect.

Recently, the deformation-dipole model has been re-examined by KUNC et a!. (1975 a, b, c) who have shown that in their version some of the above-mentioned shortcomings of the model disappear. A very satisfactory extension of the model has been given by JASWAL (1975) who has derived a complete equivalence of the deformation dipole model with the shell model in view of the microscopic theory of phonons (ZEYHER, 1975; refer to Sect. 6a).

The dipole models, in the form described in this section, need an increasing number of parameters as the complexity of the crystals investigated increases, in order to fit the experimental data within experimental error. Shell-model calculations required 11 parameters for RbI (RAUNIO, 1970), 15 for PbTe (COCHRAN, 1966), and 22 for Ti02 (TRAYLOR, 1971, KATYAR, 1974) where not even a satisfactory description was obtained. Such fits are certainly useful for calculations of specific heats, combined densities, etc.; however, the physical significance of the parameters is rather doubtful.

For the dipole models, the question arises whether or not extensions are possible which would give an improved insight into the physical nature of the force constants. If we assume the general structure of (4.24) to be approximately correct, we can guess what the potential improvements might be:

1) With respect to long-range forces, one can discuss the contribution of quadrupole and higher multipole forces. There are indications that these effects are small. Symmetry properties of multipole forces in different crystal structures have been investigated by GINTER and MYCIELSKI (1971, 1972). A discussion of quadrupole forces in alkali halides has been given by ZEYHER (1971) in the framework of an overlap treatment (Sect. 4d). He showed that the effects are small, at least for alkali halides with small atomic numbers. (For covalent crystals, see the corresponding discussion in Sect. 5).

2) There exists a contribution to the dynamical charge Zs, i.e. a charge transfer, due to the distortion of the overlap of the electronic charges, which may, for example, explain the differences of the static charges from unity in fitted dipole models. We shall discuss overlap effects in Sects. 4e and 6a.

3) The short-range deformation part, TS -1 rt (~R~ K = 1 in alkali halides, where Ro ~ K_), describes effects of a local deformability of ions of dipolar symmetry only. It is not difficult to see how a more general model theory should be designed which treats the ions in ter~s of local deform abilities with general symmetries (FISCHER et a!., 1972). A discussion is given in Sects. 4d and 4e.

We wish to note that the localization of these electronic charge density deformations at the ion lattice sites may not be as well realized in crystals with increasingly covalent character. The introduction of a "bond charge" between adjacent ions may then be preferable (see Sect. 5 c).

c) The breathing shell model. The failure of the dipole models in describing the LO mode near the L point in alkali halides shows that one needs some new mechanism not contained in simple dipolar interactions. The first attempt to introduce a type of short-range electron-ion forces other than of dipolar nature


was given by SCHRODER (1966) and NOSSLEIN and SCHRODER (1967) in the so-called breathing shell model. In this model, a compressible shell is introduced, which simulates an isotropic deformation of the electronic charge density around an ion. Only the negative ions seem to exhibit this effect in alkali halides.

Eqs. (4.18) have to be extended in the following way:

Ma/U=pU+'t"W+QV

O='t"t U +uW+QV

O=Qt U +QtW+HV,

(4.29)

where p, 't", etc. are short notations for R+ZCZ, T+ZCY, etc. The matrix Q denotes the longitudinal coupling of the breathing deformation to the nearest neighbors, while the matrix H gives the restoring force for the deformation. The dynamical matrix D(q) (see (4.19)) then becomes

D(q)=DRI+DDIP + D SPH

with the contribution of the spherical degree of freedom

DSPH(q)= _QH-IQt + (terms of higher order).

(4.30)

(4.31)

Like the Kellermann model (1 parameter) and the simple shell model (3 parameters), the breathing-shell model can also be used as a macroscopic model with its six parameters determined from the macroscopic long wavelength elastic and optic constants (CII , C12 , C44 , W TO ' wLQ, and Co or c<xJ The results of such a calculation are shown in Fig. 4.1. The agreement is nearly as good as with a nine parameter shell model fitted to neutron scattering data. One is therefore able to predict dispersion curves in crystals successfully without neutronscattering measurements as, e.g. for NaCI (SCHRODER, 1966; ALMQUIST, 1968) and, interestingly, for alkali halides with cesium-chloride structure (MAHLER, 1970; see also DAUBERT, 1971; lEX, 1971).

One feature of this model in its early version is the assumption of practically equal values of the force constants of the short-range dipolar and the breathing deformabilities. This assumption works surprisingly well for all alkali halides (SCHRODER, 1972; see also MELVIN et aI., 1970). As has been shown by SANGSTER et aI. (1970), the situation is changed as one goes to divalent compounds such as MgO. Here the breathing effect seems to be reduced as compared to that of the dipolar deformability. One difficulty of the model is that overlap effects, as mentioned in the last section, have to be simulated in an ad-hoc manner by reducing the static charge of alkali halides to values of 0.9-0.95. Furthermore, the deviation from the Cauchy relation shows no reasonable connection with the (three-body) breathing contribution to C12 (SANGSTER et aI., 1970) so that formal non-central force constants B and additional 2nd nearest neighbor forces have to be used (cf. Sect. 4a).

Recently, WEBER et aI. (1976) have investigated the lattice dynamics of rareearth chalcogenides, in particular compounds YxSml_xS, which show an interesting local change of electronic charge density due to 4f - 5d valence mixing. Since here the symmetry of the crystal does not change, it seems that a pure volume change occurs, which may be appropriately described in terms of


• (@' , \

..... : l···.--, , "

• • , ~

..... .- ( ", --. ........

t •

dipole force

"breathing" force

quadrupole force

Fig. 4.3. Symmetry of deformabilities of negative ions in alkali halides

breathing deformabilities. This point of view is supported by the corresponding decrease of C 12 (refer to GONTHERODT et aI., 1977) and has recently been confirmed by ENTEL et aI. (1979) and BILZ et al. (1979) in an analysis of the dispersion curves of SmO.75 YO.25S (MOOK et aI., 1979 a, b) (refer to Sect. 4dfJ).

d) Ionic deformabilities. ct) Deformabilities in the cubic crystals. The success of the shell model

suggests that the introduction of short-range ion deformabilities may be a useful concept in the theory of phonons. We may characterize every deformable ion in a crystal with a definite lattice site point symmetry by a set of symmetry coordinates corresponding to the different degrees of freedom in the local electron-ion interaction. We demonstrate this for the case of a negative ion in an alkali-halide (Fig. 4.3). If we restrict ourselves to nearest neighbor radial interaction of the central deformable ion to its six nearly unpolarizable neighbors, we obtain, corresponding to the six degrees of freedom (related to the displacements of the six ions in the direction of the central ion), a monopolar ("breathing") deformability of 1";.+ symmetry, a dipolar one of r15 symmetry, and a quadrupolar one of 1";."2 symmetry. This obvious reduction of the local symmetry at the lattice site of an ion which is at rest for a specific phonon can also be obtained by a rigorous symmetry analysis of corresponding phonons, e.g. those at the L point (refer to Sect. 3d). Following (4.19) and (4.30) we may now define a generalized short-range dynamical matrix R' which, in the case of cubic ionic crystals, would have the form (COWLEY, 1962; SHAM, 1969; BILZ, 1971)

(4.32a)

32

with

Phonons in insulators

DDEF ~ _" 1', S--1 1',t ~ L. r r r,

r

Sect. 4

(4.32 b)

where r denotes the different symmetry representations. It is clear that this concept, in order to be more than a convenient parametrization of the local electron-ion interaction, should exhibit some resemblance in the deformabilities of a certain ion to the structure of the outer electron shells and their excitations. From that point of view, the breathing effect of the anions in the alkali halides, although of pictorial form, is not easy to understand. A simple form of a microscopic theory (ZEYHER, 1970; KOHNER, 1970) in terms of one-particle wave functions shows that one should expect dipolar, breathing, and quadrupolar deformabilities with comparable probabilities, which is in contrast to the experimental facts.

Another possibility is to look for deformabilities at positive ions, where the central ionic potential still governs the behavior of the valence electrons. Interesting candidates are the Cu + and Ag+ ions in the corresponding halides. As is well known from the investigation of vibronic spectra of such compounds (FUSSGANGER, 1968, 1970), the valence electrons of Cu + and Ag+ ions in the filled outer dlO shells possess low-lying local excitations of quadrupolar symmetry into unoccupied s states. We therefore expect a corresponding quadrupolar deform ability of those ions to show up in the lattice vibrations of copper and silver compounds. From this point of view, an analysis of AgCl and AgBr has been given by FISCHER et al. (1972). AgCI may be compared to RbCl, which has the same structure and the same anion, but a slightly increased lattice constant, though the Rb+ ion lacks the d-shell of the Ag+ ion.

Fig. 4.4 shows the dispersion curves of both substances as measured by neutron spectroscopy. While RbCI behaves like a "normal" alkali halide and

,....,7 ~ o

N

~6 >

5

4

3

2

[OO~J

06 0.8 1.0 os 0.6 0.4 02 o 0.2 0.4

Fig. 4.4. Dispersion curves of RbCI and AgCI


can be described rather well by the breathing shell model, the dispersion curves of AgCl show some unusual features, especially the low-lying TO mode at the L point which nearly touches the T A-branch. Starting from the breathing shell model data for RbCl, FISCHER et aI. (1972) have introduced the following modifications

1) a quadrupolar deform ability (Fig. 4.3) of the silver ion (the rubidium ion can be considered to be rigid);

2) an increased Cl-CI interaction due to the decreased lattice constant (Llro/ro ~ 1/6);

3) a "valence-bond" force between the Ag+ and CI- ions corresponding to a certain amount of covalency in this compound.

The last effect is clearly indicated by the strong decrease of roo Usually one describes this type of force by bond-stretching, bond-bending, etc., forces without taking full advantage of the local site symmetry (see Sect. 5). While the radial part of this Ag-CI interaction can be taken into account by re-adjusting the values of the nearest-neighbor force constants, the lateral coupling can be described by attaching a rl~ "deformability" to the Ag+ ion (see Table 4.1) which, if one of the Cl- ions is displaced perpendicularly to the Ag-Cl line, tends to carry the four n.n. CI- ions lying in an x y plane in a type of local rotation around the Ag+ ion. The results of a nine-parameter quadrupolebreathing-shell-model are very satisfactory (FISCHER et aI., 1972). Recently, DORNER et aI. (1976) have investigated the lattice dynamics of AgBr and confirmed the prediction of FISCHER et aI. of an inversion of the transverse modes with regard to their eigenvectors near the L point (see also KANZUKI et aI., 1974; VON DER OSTEN and DORNER, 1975). This supports the idea of a quadrupolar deformability of silver which may be influenced by a hybridization of the silver d-states with the anion p-states. It is interesting to note that an extrapolation of the force constants in the sequence AgCl, AgBr, AgI leads to the prediction of an instability of the (pseudo)-T A mode at the L point in AgI (KLEPPMANN and WEBER, 1979) which is consistent with the fact that AgI does not exist in the sodium-chloride structure. Furthermore, it has been shown by KLEPPMANN and BILZ (1976a, b) that the quadrupolar deformability of the silver ion might be essential for the understanding of the high ionic conductivity of Ag+ ions in silver halides.

In spite of the practical success of this type of model calculations one might ask whether these "deformable-shell" models are more than a rather fortuitous simulation of the electron-ion interaction in lattice vibrations without any deeper significance. As we shall see in more detail in Sects. 9, 18 and 27, rather strong evidence for the presence of such local charge density deformations can be found in the interpretation of first- and second-order Raman spectra of ionic crystals. We may, therefore, try to give a rather general description of phonons in terms of local distortions of the electronic charge density by ionic displacements in order to provide a basis for our model discussions of infrared and Raman spectra in the following sections (BILZ et aI., 1974).

f3) General treatment of deformabilities. We write the harmonic potential in the following form:

(4.33)


with the "ionic" part (related to the core displacements):

cPii =1 L u(L) ifJee(L, £) u(£) =1 u ifJee u

the electronic part:

L,L'

cPee =1 L L wr(L) ifJrr(L, £) Wr (£) L,L'r

=1 L WrcPrrWr rr

and the electron-ion interaction:

cPei = 1 L (Wr . ifJre u + u ifJer wr)· r

Sect. 4

(4.34)

(4.35)

(4.36)

Here the Wr are generalized electron coordinates of symmetry type r. For example, if in a cubic crystal r = I;,5' wr(L) denotes a usual displacement vector; if r = I;,~, wr(L) describes a rotation about a principal axis through the ion labeled by L; and so on. The equations of motion include the adiabatic condition, that is, the generalized forces on the electrons must vanish:

-Mii(L)=aa(cP ) = L [ifJee(L, £) u(£)+ L ifJcr(L, £) wr (£)] u L L' r

0= a acP(L) = L [ifJre(L, £) u(£) + L ifJrr(L, £) Wr (£)] Wr L' r

(4.37)

or, in an easily understandable short notation and using the Fourier transformation of (3.24) (A ~ qj)

M W2(A) U(A) = ifJe(q) U(A) + ifJee(q) W(A)

0= ifJee(q) U(A) + ifJe(q) W(A). (4.38)

Eliminating the coordinates W from the first equation by using the second one, we obtain the dynamical matrix D(q):

MW2(A)U(A)=D(q) U(A), (4.39) where

(4.40)

If we restrict r to I;,5 and split the force constants into long- and short-range parts, we obtain the generalized (rigid) shell model. (We shall denote the force constants by cPee' cPes and cPss in this case).

When introducing further electronic degrees of freedom with the local symmetry character r, we obtain additional "deformabilities" as discussed above. As an example, we have listed the possible types of symmetries r in a crystal with NaCl structure in Table 4.1. Only n.n. radial and lateral couplings are considered. The elements of the dynamical matrix are given under the assumption that the deformabilities of the ions are isolated and not coupled to a deformability of the nearest neighbors.


Table 4.1. Symmetries of radial and lateral couplings of ionic deformabilities to nearest neighbors. D.p are the elements of the dynamical matrix. For dipolar F15 coupling, additional non-zero matrix elements exist, namely

for the radial coupling, and

for the lateral coupling (K '*' K').

Symmetries (after: Fischer et al. 1972)

Dxx(KK'lq) = -Cy-C.,

DxAK'K'lq)=2

Multipole

Radial displacements to the interconnecting lines

F i +

-t Breathing (Monopole)

Ai.

FiS- t- Dipole

Tiu

r 12 +

+ Quadrupole

E.

Lateral displacements to the interionic connecting lines

F iS +

~ Rigid rotation

Tig

r iS -

~ Dipole

Tiu

F 2S +

t:f Quadrupole (shear mode)

T2g

r 2S -

If Tetragonal deformation

T2u

D~x(KKlq) D~/KKlq)

S; SxSy

C: 0

2S; -SxSy

S;+S; -SxSy

(Cy+ Cz)2 0

S;+S; SxSy

(Cy-CY

0

-S;-S;


Fig. 4.5. Overlap polarization in the shell model. (After BILZ et aI., 1974)

It is obvious, that such a local parametrization of the theory is extremely useful, if the force constant matrices o/c' o/e and o/ec are well localized, apart from Coulomb forces.

y) Overlap polarization. The discussion, so far, does not include overlap polarization as mentioned in the foregoing section. Recently (BILZ et aI., 1975), it has been shown that this effect suggests the introduction of a positively charged shell at the lattice site of the nearly unpolarizable cation in a shell model. The argument is illustrated in Fig. 4.5. We take advantage of the fact that in many diatomic ionic crystals the center of gravity of the overlap region of the electronic charge density lies close to the center of the positive ion, as the positive ion radius is usually much smaller than that of the negative ion. Thus, the effect of the overlap polarization is best described in a simple model as being centered on the positive ion. This corresponds to the introduction of "pseudodeformabilities" for the positive ion. In the simplest approximation, the redistribution of the overlap charge has a dipole character and corresponds to a shift of positive charge in the direction opposite to the motion of the positive ion. Thus, the model contains, in addition to the parameters of a simple shell model (two n.n. force constants A and B, shell charge Y_ <0, and shell-core interaction K_ > 0 of the negative ion), the overlap polarizability, represented by a shell charge Y+ > 0 at the positive ion and a shell-core force constant K+ > O. With these six parameters fitted to neutron data, the model gives a satisfactory description of the dispersion curves of many cubic ionic crystals (BILZ et aI., 1975). It is similar to the model VI of COCHRAN et al. (1963). Their discussion assumed, however, that the positive charge simulates a deformability of the negative ion, which corresponds to the idea of the breathing shell model.

We note, that the overlap shell model not only reinterprets the physical meaning of the breathing effects as simulating to a large extent the overlap polarization, it also reduces the contribution of second-nearest neighbor forces drastically. These forces may, in fact, be negligible in many cases, since they describe anharmonic corrections, i.e. numerical changes of the phonon frequen-


cies of a few percent. This can be shown by a calculation of second n.n. forces in alkali halides using the overlap theory (Sect. 6a). Although the overlap integrals are not small, the forces induced by changes of the overlap cancel to a large extent (R. BAUER, University of Stuttgart, private communication). A very useful analysis of the overlap shell model in alkali halides has recently been given by NORGETT et al. (1977). They showed that the long wavelength elastic and optic properties may be described consistently by Born-Mayer nearest (and weak second nearest) neighbor forces for the short-range part of the potential, modified free-ion and overlap polarizabilities for the screened Coulomb forces, and weak van der Waals anion interactions. From their paper one may realize that the description of phonons in ionic crystals becomes difficult, if the free-ion and the overlap polarization are comparable as, for example, for the potassium ion in alkali halides. This might be related to the difficulties in the interpretation of Raman spectra in potassium compounds (see Sect. 18). The extremely high values for Y+ obtained in the fitting to dispersion curves of potassium halides lead to very small values for the deformability of potassium as compared to its polarizability. This indicated, possibly, a cancellation of free-ion and overlap contributions to the (effective) deformability.

b) Perovskites. The concept of ionic and overlap polarizabilities seems to work in more complex ionic crystals, too. An interesting example are the perovskites, in particular the AB0 3 oxides. A complete set of phonon dispersion curves of SrTi0 3 has been measured and analyzed by COWLEY (1964) and by STIRLING (1972, 1976). They found that a shell model gives a satisfactory description of the dispersion curves. The shell charges at the strontium lattice site were always taken as positive (corresponding to a negative deformability) in the six different models used by Stirling while the titanium shell charges are positive in three models. This is in agreement with an overlap shell model.

The evidence for a strong intra-ionic polarizability of the oxygen ion in the Raman spectra of earth alkaline oxides (refer to Sect. 18) led MIGONI et al. (1975, 1976a) to consider the effect of intra-ionic anisotropy of the oxygen polarizability on the lattice dynamics and the Raman spectra in AB0 3 perovskites. It turned out that the linear and the quartic intra-ionic polarizability is stronger. in the direction of the neighboring B ions (oxygen B-polarizability) than that in the perpendicular A-O planes (oxygen A-polarizability). While the quartic Bpolarizability gives a very satisfactory one-parameter description of the secondorder Raman spectra in KTa0 3 and SrTi0 3 (refer to Sect. 18) its thermal expectation value leads to a quasi-harmonic temperature dependence of the anisotropy which seems to trigger the ferroelectric soft-mode behavior of these incipient ferroelectrics (MIGONI et aI., 1976b). The very interesting interrelation between the lattice dynamics of soft modes in perovskites and in other crystals and the corresponding phase transitions is beyond the scope of this article. The reader is referred to the numerous and constantly increasing literature. Useful recent reviews are by SCOTT (1974); SHIRANE (1974); BLINC and ZEJts (1975); COMES and DORNER (1976); LINES and GLASS (1977), BUSSMANN-HoLDER et al. (1983); with many additional references.


e) Non-central and many-body forces and the elastic properties of crystals. During the course of our discussion so far we have used mainly two types of lattice dynamical forces:

0:) two-body formal forces as discussed in Sect. 3c, Eq. (3.27), denoted by cPap (1 I(; 11'1(;'). These forces are subject to all symmetry requirements of the crystal space group. They include, generally, non-central forces, although, for example, a pair potential between two free particles must, of course, be central due to angular momentum conservation.

/3) model forces, in particular forces connected with deformabilities, such as the breathing deformability (refer to the foregoing sections), which are generally many-body non-central forces, although they all originate eventually from twobody microscopic central Coulomb forces between nuclei and electrons.

The use of both types of forces side by side does not, in principle, lead to any difficulties as long as the interrelations between them are carefully kept in mind. Confusion may be caused by trying to introduce further restrictions on the formal force constants by starting from a specific type of interaction potential. It is clear that such a potential is subject to additional requirements, for example, equilibrium conditions, of the type (4.12), which connect the first-order derivative of a central pair potential with the central part of the lateral secondorder force constant B in a cubic diatomic crystal. This, however, does not allow for an elimination of the non-central part of B (often called B'), not even in a nearest-neighbor approximation, because the non-central part of B represents the simplest possibility to consider approximately deviations from the Cauchy relation, C12 = C44 , due to unspecified many-body forces and/or anharmonic renormalization effects which are evident from the temperature dependence of the force constants.

A more serious problem arises from the finite size of a crystal which leads to the question of whether the use of periodic boundary conditions or of the infinite size limit are appropriate methods to eliminate possible surface effects on the internal crystal forces. This question is of particular importance for long wavelength distortions of the crystal equilibrium, i.e., in the elastic limit. The calculation of the elastic constants can be done in two different ways, namely

1) by the method of long waves (corresponding to sound wave measurements),

2) by the method of homogeneous deformation (corresponding to static measurements).

The second method has the specific difficulty that, in an infinite crystal, a static strain leads to infinite displacements of the atoms as one approaches the crystal surface. One has, therefore, first to re-arrange the formulation of the crystal elastic energy in such a way that it becomes insensitive to boundary effects. Here, we mention a few essential points of the problem whereby we follow the discussion as given by M. LAX (1965a).

0:) Born-Huang conditions (BORN and HUANG, 1954). These conditions are symmetry relations between the different components of the symmetrized fourthrank elastic tensor (Born-Huang tensor)


~ _1 Clm,ns=Z(Cln,ms+ CIs,mn)

= ens, 1m (no initial stress). (4.41)

They were obtained by Born and Huang from the assumption that macroscopic and microscopic theories should lead to the same result. We note that, for central forces, the tensor C becomes totally symmetric and fulfils, in crystals where every particle has inversion symmetry, the Cauchy relations. If the initial stress is not zero it should be possible to find it from the anisotropy of the Born-Huang tensor:

(4.42)

while the initial external pressure P is related to the diagonal components

P= --t(Sl1 +S22 +S33)= --tTrS. (4.43)

This cannot be determined from the corresponding equation

(4.44)

since it defines only the traceless part of S. The Born-Huang conditions mean that certain force constants vanishing in a stress-free solid would have nonvanishing values if the initial stress is non-zero. Unfortunately, the initial stresses are generally unknown but this, in practice, does not affect the normal treatment of lattice forces in terms of formal force constants. In order to illuminate the problem we briefly discuss the case of a one-dimensional chain which has often been quoted in the literature.

/3) One-dimensional chain with nearest-neighbor interaction. Let us consider a finite one-dimensional linear chain with 2N + 1 particles and inversion symmetry at the central particle. The chain particles may be situated along the xaxis and only motion in the (x - y) plane may be admitted. Then the potential energy reads, with nearest-neighbor interactions only,

+N +N

V=t1J(1 I [ux (I)-ux (l-1)]2+t/31 I [uy(I)-uy(I-1)Y (4.45) -N -N

It is easy to show (refer to LAX, 1965a) that rotational invariance requires (starting from the end particles) /31 = ° and therefore seems to forbid the simple model in question. One then continues by introducing second-nearest neighbor forces, specified by a force constant /32' which leads to the condition /31 +4/32 =0, equivalent to bond-bending forces. One may conclude that a n.n. interaction model for transverse motion of a chain is unphysical. The infinite chain limit does not help because rotational invariance should be applied to the chain before going to this limit.

The weak point in this argument is that it treats the surface effects in an unphysical way. We should expect that near the surface the values of the forces are modified compared with those in the interior of the chain, due to relaxation, and that additional forces appear due to symmetry-breaking effects (warping, etc.). In the above example we may assume that the force constant /31 has a value P~ =l= P1 at the ends of the crystal and that bond-bending surface


forces with f3 2 (N, N - 2) =1= 0 appear which have no counterpart in the interior. The condition 4f32(N,N-2)+f3~=0 may then be fulfilled in such a way that exactly the original nearest-neighbor f3 survives in the interior of the chain. This procedure is obviously equivalent to the ad-hoc introduction of an infinite crystal or the use of periodic boundary conditions. In other words: If a nearestneighbor model can be used successfully in the explanation of experimental data (that is the real goal of most of the model treatments) it means only that the unknown surface forces are (within the limits of experimental error) consistent with the nearest-neighbor approximation in a treatment where surface effects are thrown out at the very beginning.

We note that similar arguments apply to objections against the use of Born's nearest-neighbor interaction model in the lattice dynamics of germanium and silicon (refer to COCHRAN, 1971). It seems to be well justified in the elastic limit and its breakdown at higher frequencies is not due to an (implicitly) erroneous treatment of surface forces but due to polarization effects which do not affect the long-wavelength regime (refer to Sect. 5b and 5c).

The problem of rotational invariance can be considered from a more general point of view. If one applies this invariance condition to the complete lattice potential including all anharmonic expansion terms, one obtains a set of equations where always a given (say second) order term of the potential is coupled to the next higher (say cubic) order term. That means that rotational invariance cannot be fulfilled strictly in the harmonic approximation (refer to LUDWIG, 1967; LEIBFRIED, 1968). In practice, we may usually neglect those anharmonic corrections at sufficiently low temperatures or consider them by a renormalization of the harmonic force constants. Purely harmonic lattice dynamics is therefore not possible, but only in a quasi- or pseudo-harmonic approximation. (For a similar argument with respect to homogeneous deformation, see LUDWIG, 1967; LEIBFRIED, 1968. Here a purely harmonic crystal would become unstable against a volume decrease.)

y) Central and non-central forces. The conservation of angular momentum restricts pair potentials of free particles to central potentials. That means, that formal two-body, non-central forces in a crystal originate from third- or higher-order physical many-body forces. In an ionic crystal, the equilibrium condition (4.12), describes the balance between the central Coulomb forces and the central part of the short-range forces only. The non-central lateral nearest or higher neighbor force constants, BNC , B~c, etc., are disposable parameters in every fit model. Their practical importance lies in the fact that deviations from the Cauchy relation ('Cauchy violation') may be described to a first approximation by a non-central BNC • For example, in an alkali halide crystal we have

C12-C44OC-2BNC-.... (4.46)

Since we have seen in the foregoing sections that many-body forces due to breathing or quadrupolar deformabilities lead to a Cauchy violation, we prefer to say that a use of noncentral forces in a fit procedure corresponds to the consideration of unspecified many-body forces in a global form.

It is sometimes felt that this pragmatic attitude is not satisfactory particularly in a treatment where the lattice potential is constructed from pair-,


three-body, and so on, forces which are explicitly invariant against local translations and rotations in the same way in which the forces of a diatomic or three-atomic molecule have to be invariant under local symmetry operations. If, in addition, the forces in a crystal can be determined by a straightforward extrapolation from molecular forces (see, for example, the discussion of valence forces in diamond, Sect. 5d) which indicates that lattice forces may be determinated from rather small units in the crystal, such as bonds or tetrahedra, one is tempted to dispute the additional use of formal forces which are not subject to local invariance conditions. The situation is reminiscent of the discussion of surface forces in the foregoing paragraph. If it turns out in an actual calculation that a model fit is definitely improved by including non-central formal forces, it does not mean that one has obtained the better fit at the expense of using an unphysical model (as it is sometimes stated in the literature). It rather indicates that the model is incomplete with respect to the number and type of 'local' forces and that extensions of the model are necessary. While this should eventually be the best way to improve a calculation, it might well be an impossible task, for the time being. Then, a pragmatic use of formal force constants seems to be better than the renunciation of a satisfactory fit.

One should note that effects exist which make it difficult to believe that formal force constants can ever be completely avoided in an actual calculation. For example, the transverse acoustic branches in germanium exhibit an anomalous temperature dependence of the frequencies near the X point in the B.Z. (BROCKHOUSE, 1960). It has been shown by JEX (1969) that the anharmonicity of the lattice potential, which is connected with this effect, also leads to a frequency-dependent renormalization of the transverse acoustic frequencies of several percent. In the quasi-harmonic treatment of the dispersion curves of germanium with effective model parameters this detuning has to be simulated by a re-adjustment of the parameters, or by considering further force constants. As this discussion shows, an analysis of lattice dynamics which considers all effects of importance and the present possibilities of taking them into account in an actual calculation leads to the conClusion that the use of (at least a few) formal (central or non-central) force constants in model calculations is well justified. The same argument holds for a discussion of non-linear potential expansion parameters, such as infrared or Raman parameters in forthcoming sections.

b) Many-body forces and the Cauchy relation. In cubic crystals the Cauchy relation is usually not satisfied. This is, to a certain extent, the reason for the introduction of many-body forces in the lattice dynamics of cubic crystals in order to obtain a more microscopic picture of the formal non-central two-body force constants discussed in the last paragraph.

BASU and SENGUPTA (1974) have given a detailed discussion of the interrelation between many- (mainly three-) body forces in existing theories and the violation of the Cauchy relation. We refer to their results, but include the additional aspects of the influence of deformabilities from the foregoing sections. A point of importance is the consistency of the effects on the elastic constants with those on short-wavelength phonons, in particular near the L point of the Brillouin zone.


To be specific, we discuss the case of alkali halides with rocksalt structure. Then, the elastic constants and moduli read as follows (Cik =(2rJ/e)2 C ik),

Cll = -5.11(Z +AZ)2+A+A' +B'- CB +2CQ

C12 = 1.39(Z +AZ)2 +t(A' _B')-BNC- CB - CQ

C44 = 1.39(Z + AZ)2 +t(A' - B') + BNC

Cll +2C12 = -2.33(Z +AZ)2+A+t(3A' +B')-2BNC-3 CB

- - 2 1 I I Cll - C12 = -6.50(Z +AZ) +A+z(A +3B )+BNC+3 CQ

C12 - C44 = -2BNC- CB - CQ .

Here the formal quantities are 1) the valency Z (= 1, for free ions), 2) A, A', B', n.n. and 2nd n.n. force constants, 3) BNC' a non-central n.n. force constant.

The many-body parameters are

(4.47)

(4.48)

(4.49)

(4.50)

(4.51)

(4.52)

1) AZ, a three-body overlap correction to the static charge (refer to Sect. 6 a),

2) CB , CQ, many-body corrections due to breathing or quadrupolar deformabilities (refer to Sects. 4c and 4d).

We note that the (long- and short-range) dipolar deformations in the shell model do not affect the elastic constants due to inversion symmetry of the ion lattice sites. Let us first consider the long-range effects. LANDSHOFF (1936), LOWDIN (1948), and LUNDQVIST (1955, 1956, 1957) have developed an overlap theory which is outlined in Sect. 6a. With respect to the present problem the result of a nearest-neighbor three-body approximation in the overlap are two long-range corrections to the rigid-ion terms (SINGH and VERMA, 1969; ZEYHER,1971),

1) a correction AZ of the first static charge Z (Eq. (6.1), generally of a few percent to lower values) which may easily be incorporated by treating Z as an adjustable parameter close to the free-ion static charge. The changes of cohesive energy are then only a few percent and still consistent with experimental values.

2) More important is the appearance of a long-range breathing term (refer to (6.1) and (6.2))

(4.53)

which leads to a Cauchy 'deviation' C 12 - C 44 < 0, since the overlap derivative f' < 0. We note, however, that if we replace f' by the more general asymmetryparameter derivative C Eq. (6.2), both signs are allowed for CB • In fact, since in 30 % of the alkali halides (including NaI with the strongest breathing effect at the L point) C12 - C44 >0, experimentally, the sign of the breathing contribution to the Cauchy violation is not consistent with the lowering of the LO mode at the L point, which requires C B > 0, i.e. C 12 - C 44 < 0. The good fits obtained by SINGH and VERMA (1969, 1970, 1972) in the so-called three-body shell model, as well as in the short-range breathing-shell model (SCHRbDER, 1966; NOSSLEIN and SCHRbDER, 1968) or in the slightly modified models by


MERLIN et aI. (1969) and BASU and SENGUPTA (1969) calculated for crystals with C12 - C44 >0 can only be considered as fortuitous. Generally, secondnearest neighbor (A', B') forces and a non-central BNC have to be used to obtain agreement with both elastic and zone boundary phonons. Therefore it has been suggested to replace these off-diagonal three-body breathing forces, located usually at the more polarizable negative anion, by a better converging overlap dipole polarization located at the positive cation site (BILZ et aI., 1975) which contains diagonal three-body forces only like the simple shell model which do not affect the Cauchy relation. We come back to this point at the end of this section.

While the three-body terms corresponding to breathing deformations seem to play a role in the mixed-valence compounds (BILZ et aI., 1979) there is evidence for many-body terms CQ connected with quadrupolar deformations of the d-shell of Ag+ ions in silver halides (refer to Sect. 4c). Interestingly, there exists a similar trend in the elastic constants of the pure metals, Cu, Ag, Au, (SIMMONS and WANG, 1974) indicating an increasing importance of quadrupolar deformations of the filled d-shell of these atoms.

The success of the introduction of many-body forces or ionic deformabilities for the explanation of the Cauchy violation seems to be a very moderate one. We note that the shear modulus C44 Eq. (4.49), is not affected by any of the many-body terms discussed so far, while the appearance of BNC in the equation for C44 indicates that other many-body forces may exist which have so far been neglected. In the specific case of alkali halides this is expressed by the fact that the above-mentioned overlap shell model (BILZ et aI., 1975) which contains dipolar deformabilities only and rather weak second-nearest neighbor interactions, requires the use of the non-central BNC to account for the Cauchy violation. It is hoped that a more accurate treatment of overlap effects (ZEYHER, 1971; GLISS et aI., 1970) will lead to a better understanding of this problem. An obvious possibility is the consideration of lateral nearest-neighbor forces of I;~ symmetry (refer to Table 4.1). For covalent crystals the introduction of bond charges leads to a successful and pictorial description of anomalies of the transverse acoustic branches along the (100) and (111) branches which are intimately related to the shear modulus C44 (refer to Sect. 5c).

For similar difficulties to explain the Cauchy deviations of alkali halides in terms of interatomic low-energy charge-transfer matrix elements see HARRISON (1980, p. 463).

y) Molecular crystals. In molecular crystals the point-like constituents of the lattice are replaced by molecules with non-adiabatic internal degrees of freedom. The lattice vibrations can (to a certain approximation) be devided up into internal (molecule-like) and external (crystal-like) modes (refer to VENKAT ARAMAN and SAHNI (1970)). The lattice dynamics of molecular systems may be described in a way analogous to crystals with point-like lattice particles. In particular, the interrelation of the conditions for equilibrium and stability with the (invariant) elastic and other force constants is similar to the corresponding relations in an atomic or ionic lattice. The reader is refered to FALTER and LUDWIG (1971), GEICK (1978), and references therein.


5. Covalent crystals

a) Formal force constants and general properties. In an analysis of the inelastic neutron-scattering data of germanium HERMAN (1959) showed that, in order to obtain a satisfactory description of the dispersion curves, at least fifteen Born-von Karmtm force constants (that is out to fifth-nearest neighbors) have to be introduced. This long-range behavior of the effective ion-ion forces is somewhat surprising since Born's relation (BORN and HUANG, 1954) for the elastic constants indicates n.n. forces only, at least in the long-wavelength acoustic regime. Moreover, a sum rule (BLACKMAN, 1942; BROUT, 1958; ROSENSTOCK, 1963) is well satisfied by the experimental dispersion curves in Ge, Si, O(-Sn, and many III-V compounds, which indicates that the nonelectrostatic part of the interatomic forces must be of short range (refer to Sect. 5 f). On the other hand, the very low coordination number of four in these crystals exhibits immediately the strong anisotropy of the forces in the diamond structure and makes a treatment in terms of more or less central potentials doubtful from the very beginning.

Two features seem to be important in the discussion of phonons and related properties in covalent crystals. First, these crystals, in particular diamond, behave in many respects like huge molecules whose dynamical properties can be derived from those of very small units. Thus, for example, the C-C interaction as known from organic molecules can be used in order to determine the interaction in diamond. Quantum-mechanically this originates from the Sp3

hybridization and leads to the concept of valence forces which will be discussed in Sect. 5 d. The second interesting aspect is the fact that the crystals of the diamond structure may be divided into two classes. The first - class, which contains essentially diamond itself, shows a weak crystal polarizability (e oo = 4 in diamond) and high values of the elastic constants, while the crystals of the second class (Si, Ge, and O(-Sn) exhibit high values of e (11, 16, and 24, respectively) and low values of the transverse elastic constant C44 . In addition, they have a rather flat T A branch with anomalous anharmonic properties (see lEX, 1970). In Fig. 5.1, the dispersion curves of diamond and silicon in the main symmetry directions are compared. The dispersion curves of germanium and grey tin are very similar to those of silicon. This can be shown by scaling the frequencies with the Raman frequency (refer to SINHA, 1973, Fig. 10).

b) Dipole models. For the analysis of infrared and Raman spectra of covalent crystals, a treatment of their phonons which is related in a natural way to concepts such as polarizability, effective charges, etc., is desirable. Therefore, the dipole models which have been applied successfully to ionic crystals might be examined for their usefulness for these crystals. It cannot be expected, however, that the success of these models will be as striking as for alkali halides. The main reason is that the electronic charge density is not localized within separated spherical volumes around the ions but accumulates partially into rather isolated "bond charges" between the ions, due to the hybridization of sand p electrons. An example is the charge distribution in GaAs as calculated by WALTER and COHEN (1971). A small asymmetry, as

Sect. 5 Covalent crystals 45

1.0 LO

SILICON

Z (100)

« ~ « 0.5 0:: 3 ""-3 TA

a

1.0 LO

TO

Z DIAMOND SILICON « (111) (111) ~ « : 0.5 ""-3 TA

TA

a 0.5 1.0 1.0 0.5 a q/qmax- -q/qmax

Fig. 5.1. Comparison of dispersion curves in diamond and silicon. (After WARREN et aI., 1965)

compared with the distribution expected for silicon, originates from the weak ionicity of GaAs.

MASHKEVICH and TOLPYGO (1957) drew attention to the importance of induced dipole forces in covalent crystals. COCHRAN (1959) first used the shell model for germanium and showed that a live-parameter treatment gives surprisingly good results in the main symmetry directions. Since the three elastic constants (Cll , C12 , and C44), the Raman frequency OJR and the dielectric constant 8 are available as macroscopic entities, a successful five-parameter model with values of the parameters derived solely from the macroscopic data could be considered a very important step in the lattice dynamics of covalent crystals. Cochran's treatment begins with a dynamical matrix given by (4.19), but puts the static charge Z equal to zero:

D(q)=R- T(S+ YCy)-l Tt. (5.1)

In the nearest-neighbor approximation, R contains two parameters. Furthermore, T and S (see (4.20)) are assumed to be proportional to R. Since one parameter turns out to be redundant, only five parameters remain, including the shell charge Y and the shell-core spring K. Then agreement between experimental data and theoretical values fitted to them is within 10 %. One needs eleven parameters in an extended shell model to obtain agreement within experimental error (typically of 2-5 %) (DOLLING and COWLEY, 1966). Even then, the agree-


mentis poor at critical points like W, q =(11-0) n/ro, which are important in the one- and two-phonon density of states. This leads to clear deviations between calculated and measured infrared spectra (see Sect. 16).

The clue to the relative success of Cochran's rather simple model can be found in the description of the crystal polarizability. Since in the crystal under consideration the static charge Z is zero, there exists no macroscopic field at long wavelengths,and the three optic branches are degenerate. Nevertheless, the Raman frequency is lowered by the short-range, i.e., the mechanical part of the crystal polarizability, like the center frequency in alkali halides (see (4.23)). In the latter case the short-range electron-ion interaction is completely localized; hence there exists only a shell-core spring constant between the shell and the core of the same atom. Consequently the electronic polarizability, oc el , Eq. (4.28), becomes nearly a constant for every single ion. Cochran's model, on the contrary, contains a coupling between shells and cores of neighboring ions which may be interpreted by q-dependent polarizabilities depending on how much the shells of neighboring ions are out of phase. Interestingly enough, such non-local electron-ion couplings have been used successfully for other highly polarizable materials like thallous bromide (E.R. COWLEY and OKAZAKI, 1967) and silver chloride (VUA YARAGHAVAN et aI., 1970). It seems that this indicates the need to consider other types of ionic deformabilities (for example a quadrupolar deformability, as in AgCl, discussed in Sect. 4e), or to extend the concept of individual and separable ionic polarizabilities to a more general description of crystal polarizability. The second idea is very plausible in view of the charge distribution in covalent crystals.

In alkali halides, the introduction of a "breathing" deformability of negative ions drastically reduces the number of parameters. The same is true, somewhat surprisingly, for the covalent crystals, as has been shown by KRESS (1972). His model can be considered as an extension of Cochran's model in the same way as Schroder's breathing shell model for alkali halides is an extension of the simple shell model. It uses the concept of q-dependent deformabilities as discussed above. Although such a description is probably not the ultimate answer to the problem of lattice dynamics of covalent crystals, it has the great advantage, compared with the valence-force approach, of giving a physical description of polarizabilities and deformabilities which is important for an understanding of infrared and Raman spectra (Sect. 17).

c) Bond-charge models. A better description of the electronic charge accumulation between the ions can be expected from the "bond-charge" model, first introduced by WARREN (1965) and extended by MARTIN (1968, 1969) and PHILLIPS (1968). In this model, the bond charge is represented by a point charge at the midpoint between neighboring ions. The nearest-neighbor forces can be derived from a diagonal dielectric constant e in a pseudo potential calculation (see Sect. 6) which corresponds to the idealization of the electronic charge density to a homogeneous electron gas. Since this leads to central forces between nearest neighbors only, but retains shear instability of the crystal, the bond charges are used to describe the non-diagonal part of e which is required by the acoustic sum rule (Sect. 6) for obtaining stable transverse acoustic vibra-


tions. As a result, the elastic constant C 44' which determines the transverse sound velocity along the main axis, is obtained as a small difference between large Coulomb and short-range forces.

Martin's zero- or one-parameter calculation gives fairly good agreement in view of its simplicity. SINHA et al. (1971) generalized the treatment by replacing the point bond charges by a uniform charge distribution inside a sphere of adjustable radius R. In principle, these spheres can be situated between the ions as in the bond-charge model, or at the ion lattice sites as in a generalized shell model. The result of a calculation with the latter model for silicon with one adjustable parameter (R) gives good results. Since SINHA et al. are using a parametrized form of the microscopic theory (Sect. 6), the success of their calculation shows clearly that a "first principles" calculation of phonons in insulators might be possible in the near future.

Nevertheless, a model theory of simpler structure still seems to be necessary in order to obtain a basis for the discussion of infrared and Raman data. As has been discussed by KRESS (1972), the simple bond-charge model overestimates the long-range Coulomb forces and should therefore be replaced by a 'bonddipole' model, where the bond charge is supposed to be screened in the equilibrium configuration and dipoles are induced at the midpoint between the ions with their strength proportional to the relative shell-core displacements. Such a bond-dipole shell model with six parameters leads to very good results.

The most promising approach, for the time being, seems to be a recent extension of the bond-charge model by WEBER (1974, 1977) and WEBER et al. (1974). Here, one treats the bond charge as a point charge which is not fixed to the mid-point between neighboring ions but always takes up the proper adiabatic force-free position in the displaced ion configuration. A four-parameter model gives a very satisfactory description of the dispersion curves of diamond and its homologues. Since we use this model in our discussion of the infrared and Raman spectra of covalent crystals (Sects. 16 and 19) we describe its general features in Fig. 5.2. It should be noted that the model, while sufficient for describing dispersion curves, fails to reproduce the complete crystal polarizability. Probably, a further extension of the model including a charge transfer or a charge form factor could remove this failure (see also Sect. 5e).

Fig.5.2. Bond-charge model. IPi_i(r), central potential between nearest-neighbor ions. IPi_b(r), central potential between ions and bond charges. Yrc, bond-bending potential (Keating) between neighboring bonds. 2Z, - Z, values of ion and bond charges which determine the Coulomb forces.

(After Go et aI., 1975)


d) Valence force fields. In the theory of normal vibrations of molecules, the potential energy of the molecule is often given in terms of internal displacement coordinates, such as changes of bond distances, bond angles, etc. (WILSON et aI., 1955). The corresponding interatomic forces originate from the deformations of the electronic charge density which is highly directional due to a hybridization of the electronic orbitals.

The covalent crystals such as diamond can be considered merely like very large molecules and should, therefore, be describable by an extension of the molecular methods to almost infinite systems. A very good basis for finding the appropriate set of valence force field constants in the crystals should therefore be obtainable by investigation of, say, organic molecules with similar covalent bonds. In fact, the least-square analysis of branched paraffins by SCHACHTSCHNEIDER and SNYDER (1963) has been used successfully for the description of lattice vibrations in diamond, silicon, germanium and grey tin by MCMuRRAY et al. (1967; 1972) and by TuBINO et al. (1972). The six valence-force field parameters are shown and their values for the different crystals are given in Table 5.1.

It is very impressive to note that the model originating from an analysis of carbon bonds works equally well for Si, Ge and even a-Sn. The region of greatest deviation between theory and experiment is near the L point in the Brillouin zone, as in Cochran's five-parameter model. The valence-force-field potential does not contain any Coulomb interactions, thus supporting the view taken in KRESS' paper (1972) that the interacting electronic and ionic parts of charge distributions are very well screened. The qualitative difference between diamond and the other crystals is expressed by the strong decrease of the relative strength

Table 5.1. Valence-force field parameters for diamond-like crystals. (After TuBINO et aI., 1972.) Units: stretching and streching-streching force constants are expressed in mdynes/A, bendings and

bending-bending in mdyne· A/rad2, streching-bending in mdyne/rad

Diamond Silicon Germanium Gray tin

(/"; a(<pi) <Pi a(<Pi) <Pi a(<Pi) <Pi a(<p)

Kc 3.831 ±0.023 1.469 ±0.031 1.330 ±0.021 0.974 ±0.015 HA 0.872 ±O.121 0.164 ±0.01l 0.142 ±0.018 0.088 ±0.016 FR 0.164 ±0.017 0.052 ±0.008 0.031 ±0.007 0.098 ±0.005 FRA 0.392 ±0.012 0.073 ±0.008 0.096 ±0.010 0.045 ±0.010 FA' -0.015 ±0.01O -0.025 ±0.013 -0.026 ±0.010 0.024 ±0.017 FA" 0.173 ±0.043 0.132 ±0.021 0.134 ±0.027 0.109 ±0.019

C C C c

i J\A /C,

I \~ C/}c ~C c/j ~C /.:C~ * C-C C C C-CCA A C '-:\

C C C C

Kc HA FR FRA FA' FA"


of the bond-bending force constant H A (Table 5.1) from a value of about 25 % of the stretching force constant Kc in diamond to values of about 10 % in the other crystals.

In spite of its usefulness for the description of phonons in covalent crystals, it seems that the valence-force potential model lacks both a clear relation to the dielectric properties of the crystals and a link with more ionic crystals which gives the shell model its strong appeal. One might guess that the different valence-force constants as described in Table 5.1 show certain relationships to the deformabilities described in Sect. 4.

Here, a recent treatment of the lattice dynamics of covalent crystals by V ASILEV et al. (1971) is of interest. These authors describe the "rigidity" of the tetrahedron of directional bonds originating from the hybridization of atomic wave functions by attaching a tetrahedral structure of four "rays" to every single atom. The structures between neighboring atoms are then coupled in such a way as to lead to adiabatic local rotations. Unfortunately, the paper by V ASILEV et al. contains a mistake which makes their four-parameter calculations invalid for the transverse optic branches while giving the expected good description of the flat transverse acoustic branches (W. KRESS, private communication).

e) Crystals of partially ionic and partially covalent character. The covalent structure of diamond-type crystals is essentially retained in crystals like GaAs, which is a typical example of a class of crystals with cubic zinc blende or wurtzite structure, mainly represented by the III - V and II -VI compounds. The strong similarity between this class of crystals and the purely covalent ones originates from the fact that every single ion in these compounds is still surrounded by four nearest neighbors in a tetrahedral configuration. On the other hand, differences appear due to the lack of inversion symmetry in compounds like GaAs. The effect on the dispersion curves of the III - V compounds is often not dramatic: it leads to a very small splitting of the Raman frequency into longitudinal and a two-fold degenerate transverse optic mode while retaining essentially all the other features of the dispersion curves of the homopolar crystals. On the other hand, the changes are much more important for the infrared and Raman spectra, as will be discussed in detail in Sect. 16.

From this discussion, it seems clear that two approaches to the lattice vibrations can be expected to be more or less successful in the case of partially ionic and covalent crystals. The first would be a type of dipole model, like the shell model and its generalizations, and the second a mixture of a valence-force field treatment with a (simpler) dipole model.

In practice both types of models have been used. For the highly polarizable GaAs (DOLLING and WAUGH, 1965) and GaP (YARNELL et aI., 1968), a fourteen-parameter shell-model treatment gives a good description of the measured dispersion curves, but the physical meaning of most of the parameters used in the fitting procedure is unclear. For ZnS, which is more ionic, a sevenparameter shell model is still rather successful (BERGS AM, 1971).

A mixture of valence-force constants or generalized short-range forces with the deformation dipole model has been used by NUSIMOVICI et ai. (1967, 1970)


02 01. 06 Q8 10 08 06 01. 02 0 OJ Q2 03 010 05 REDUCED WAVE VECTOR

Sect. 5

Fig.5.3. Phonon dispersion-curves for GaAs. The experimental data from DOLLING and WAUGH

(1965) are represented by open triangles. Solid lines show the calculated results of a six-parameter bond charge modeL (After RUSTAGI and WEBER, 1976)

for the description of CdS with wurtzite structure and by KUNC et aI. (1970, 1975a, b, c) for the zinc-blende structure compounds like ZnSe. This deformable bond model needs at least eleven parameters in order to describe the measured dispersion curves fairly well.

Very promising, at first glance, seems to be an extension of WEBER'S version of the bond-charge model (WEBER, 1974; refer to Fig. 5.2) which has recently been applied to GaAs (WEBER et aI., 1974). Figure 5.3 shows the result of a five-parameter calculation where the only additional parameter as compared with silicon is the charge asymmetry between the Ga + and the As - ion. Unfortunately, a further extension of the model to II-VI compounds does not seem to be possible since the asymmetry of the bond charge position becomes too strong (RUST AGI, private communication).

A successful combination of the overlap shell model (see Sect. 4c) with the valence-force model has recently been applied to II-VI and III-V compounds (KUNC and BILZ, 1975, 1976). The appealing aspect of this approach is that it not only gives a satisfactory description of the dispersion curves of these materials but also provides an understanding of the second-order Raman spectra in these crystals in terms of intra-ionic polarizabilities of the ions (see Sect. 19).

f) Sum rule of lattice vibrations. At the end of this discussion of model theories, a particularly interesting aspect of the situation may be emphasized. It is the fact that the covalent crystals (with the exception of diamond) and many III-V and II-VI compounds follow to a good approximation a rule which states that the trace of the dynamical matrix DO, Eq. (3.34), is a constant independent of q, that is,

6

f(q)::Tr DO(q)= L w 2(qj)=const. j= 1

(5.2)

This sume rule has been obtained by BLACKMAN (1942) and by BROUT (1958) for the case of a rigid-ion crystal with nearest-neighbor short-range interactions only (Kellermann model; see Sect. 4a). Using (3.34), it can easily be proved that

Sect. 6 Microscopic theory, models, and macroscopic quantities 51

electrostatic Coulomb forces do not contribute to f(q) since the Laplacian of the Coulomb potential equals zero, that is

LJ tjJc = L (tjJ~': + tjJ~~i+ tjJ~~i) = 0, (5.3) i

where the summation index i extends over all spheres of neighbors. The nonvanishing term in (5.2) comes from the nearest-neighbor short-range forces which give

( 1 1) 9vo f(q)= M+ + M_ r~fJ (5.4)

where fJ is the compressibility of the crystal (refer to 4.13)). This sum rule is not fulfilled in ionic crystals like the alkali halides where

f(q) usually is a function of q with its values at the zone boundaries 15-20 % higher than the value at r. However, crystals like silicon, germanium, gallium arsenide, etc., fulfil the sum rule more or less within the limits of experimental error. It seems that a physical basis for the sum rule is not easily found. In any case, the q-dependence of f(q) can be used to find the range of non-electrostatic forces in crystals as has been shown by ROSENSTOCK (1965, 1973). It turns out that in crystals like TlBr these forces extend to fifth-nearest neighbors (E.R. COWLEY and OKAZAKI, 1967) while in diamond only radial central secondnearest-neighbor interactions are required to describe the q-dependence of f(q) (ROSENSTOCK, 1963).

The fact that a great variety of crystals with diamond, zinc blende, and wurtzite structure obey the sum rule leads to the conjecture that the model theory as described in the foregoing paragraphs still lacks an important feature of the lattice potential. The shell model does not give an indication towards the understanding of this rule since it does not fulfil the sum rule apart from accidental numerical relations between the force constants.

The problem has been re-analyzed quite recently in terms of the bond-charge model (BILZ, unpublished). It seems that this model provides an understanding of the validity of the sum rule as determined by a sufficient localization of the electron-ion interaction, on the one hand, and a sufficiently high polarizability of the crystal on the other hand which screens the indirect (electron-induced) ion-ion forces nearly completely to within the nearest-neighbor distance.

6. Microscopic theory, models, and macroscopic quantities. In the previous sections lattice vibrations in insulators have been discussed in the framework of a formal description, i.e. by using (solely symmetry-restricted) force constants of the effective ion-ion interaction, or by model theories which give a pseudoclassical treatment of the adiabatic part of the electron-ion interaction. In this section we are going to discuss the existing microscopic theories of phonons in insulators and their relations to the parameters of the phenomenological formal or model theories. For recent reviews the reader is referred to papers by SHAM and by BILZ et al. in: HORTON and MARADUDIN (1974), by SINHA (1974) and by HANKE (1978).

In microscopic theories one begins with the Coulomb electron-electron and electron-ion interactions and eventually derives the dispersion relations by some


approximation to the many-body Hamiltonian. The essential assumption for the applicability of such theories is the definition of "ions" (instead of the bare nuclei) which include the tightly bound electrons. The Coulomb potential of the ions then has to be replaced by a pseudopotential which takes account (usually in a local approximation) of the exchange and correlation effects of the valence electrons with the core electrons. Then, the adiabatic electronic response to the displacement of the ions is determined with the help of a response function which eventually leads to a description in terms of the effective interionic forces in the crystal.

The microscopic theory of phonons may, for the time being, be divided into two different approaches. The first one describes the interatomic forces, except for the long-range Coulomb forces, as originating from the overlap of mainly the nearest and second-nearest neighbor charge densities. In its simplest version it leads in a natural way to a modified rigid-ion model (LANDSHOFF, 1936; LOWDIN, 1948; LUNDQVIST, 1953, 1954, 1956). We discuss this approach and more recent developments in the next section. A second form of the microscopic theory is oriented towards the description of the crystal as a dielectric medium in which the dynamical properties can be described by the dielectric response of the electronic charge density to the ionic displacements in the lattice (Sect. 6b). Finally, we discuss the relationship between these theories and some widely used phenomenological quantities, such as effective charges, bonds, covalency etc. (Sect. 6c).

a) Overlap theory. In Born's calculation of the cohesive energy of alkali halides, the overlap of the electronic charge densities between the ions is partially taken into account by means of short-range repulsive forces (BornMayer potentials). A second overlap effect is the modification of the long-range ionic Coulomb interaction. This effect has been treated quantum-mechanically by WWDIN (1948) in the Heitler-London approximation (HLA). Since one knows that the energy of ionic molecules in the limit of large distances between the ions is correctly given by the HLA, one might expect that the cohesive energy of weakly-overlapping electronic shells could be treated successfully in the HLA. As a consequence, the lattice vibrations of such crystals should also be describable rather well in the framework of the HLA.

LUNDQVIST (1953, 1954, 1956) has investigated the effect of "overlap forces" in the long-wavelength limit. The simplest approximation beyond the rigid"ion treatment is a description of three-body effects which modify the elastic constants leading to a violation of the Cauchy relation, C 12 = C 44' Furthermore, the static ("Born") charge Z is screened to an effective ("Lundqvist") charge ZL.

To understand the problem in more detail, we have to consider the approximations employed in the calculations. So far, the HLA has only been used in its conventional form, where the electrons are assumed to stay in their static equilibrium (approximately free-ion) ground states during the displacement of the ions. Therefore, polarization of the ions by excitation of electrons is not taken into account. We may therefore call such a treatment a rigid-overlap approximation (ROA) (GLISS et aI., 1971). The ROA should not be confused


with more formal treatments using orthogonal equilibrium wave functions. In these latter treatments, the effect of changing the overlap as given in the ROA has to be described by virtual excitations of the electrons from their equilibrium ground state. Convergence of corresponding calculations is usually poor. Since the ROA gives very good results for the lattice constants and the cohesive energy in alkali halides (ABARENKOW, 1970) we might expect that the ROA is perhaps the best manybody ad-hoc approach to a "rigid-ion" crystal by using nonorthogonal one-particle wave functions.

Let us first summarize the results of the ROA. Calculations of the elastic constants and of the deviations from the Cauchy relation and of ZL have been done by LOWDIN (1956); LUNDQVIST (1953, 1954, 1956); VALLIN et al. (1967); ABARENKOW (1970); and ZEYHER (1971).

In the limit q -+0 and using nearest-neighbor overlap forces in Lowdin's S2_ approximation, we obtain for the (generally q-dependent rigid overlap) charge tensor (ZEYHER, 1971) the expression

(6.1 )

The static charge Z + is changed by a factor which contains the asymmetry parameter

,=(J(f+ - f- -(g+ -g J). (6.2)

Here, (J is the sum of the squared overlap integrals between nearest neighbors, while f + and f _ are approximately the relative ionic radii of cations and anions (f + + f _ = 1). g + and L denote some first-order moments of the charge distribution centered at the ions, and ,~ is the equilibrium value of the first derivative of , with respect to a displacement of the ion in the IX-direction.

The first correction term in (6.1) corresponds to an increase of the static charge to higher values proportional to the difference in ionic radii. This charge renormalization is a few per cent in LiF but practically zero in KF, where cations and anions are of nearly the same size. The second correction term leads to a decrease of the static charge due to the variation of the parameter , when the ion under consideration moves in the IX-direction. This term usually overcompensates the first term, giving mean charges of about 0.95 e for LIF in the long-wavelength limit.

The consequence of these long-range overlap corrections are: 1) The deviation of the elastic constants from the Cauchy relation, C 12 = C44 ,

is qualitatively described with the right order of magnitude and the correct sign. However, the quantitative agreement is still unsatisfactory since other shortrange many-body forces and anharmonic corrections are not taken into account.

2) The overlap screening of the static charge to values smaller than 1.0e may explain the improvement of dipole models when the static charge is used as a variable parameter. In alkali halides this gives values near 0.95 e for LiF and even lower values for NaI (0.9 e) and RbI (0.85 e) (SCHRODER, 1972). In this way, the dipole models partially take into account the many-body overlap effects.

The charge transfer due to the overlap of the electronic shells was accounted for by DICK (1962) and MARSTON and DICK (1967) in their exchange-charge model. Unfortunately, they developed the model no further after they found that the breakdown of the shell model for LO at the L point in the B.Z. in alkali


halides (cf. Fig. 4.1) was not eliminated in the exchange-charge model. A similar failure was observed by ZEYHER (1971) in his quantum-mechanical treatment of overlap charges in the ROA.

In this respect the results obtained by VERMA and SINGH (1969a, 1969b, 1970) are debatable. They used only aU + - f _) and its derivative as adjustable parameters, neglecting g + and g _ and some other q-dependent terms in the expression for Z(q) (ZEYHER, 1971). It seems that these shortcomings have not been removed in recent papers (BASU and SENGUPTA, 1974).

The ROA discussed here is quite in the spirit of the adiabatic theory which needs only ground-state electronic wave-functions in order to describe the lattice potential. As the discussions of this section suggest, the free-ion wavefunctions used in the Heitler-London calculations of overlap effects (ZEYHER, 1971) may be a useful first approximation for the equilibrium configuration but not for a displaced arrangement. In this case one would need an overlap polarization part in addition to the rigid-overlap part in the dynamical matrix (GLISS et aI., 1971), which means that in a second-order approximation the electrons do not stay in their approximate free-ion ground state during the ionic displacements but are virtually excited leading to a deformation of the electronic charge. A better approximation for the crystal ground state should give results analogous to those obtained from the dipole models but on a more microscopic basis ..

A straightforward approach to a proper calculation of the ground-state wave function has been proposed by ADAMS (1961, 1962) and GILBERT (1964). As in a paper by SINHA (1973), the deformation part is deduced from a pseudopotential which should ensure a rapid convergence of the perturbative sums. Recently, ZEYHER (1975) used this approach and succeeded in calculating the dispersion curves of LiD without any adjustable parameters. His results agree with experiment to within 10 % on the average. He demonstrated clearly the usefulness of highly localized non-orthogonal wave-functions for the deformation part. Here, a small parameter appears in a natural way, namely the overlap between occupied orbitals of different ions. The effect of ionic polarizabilities is considered in this treatment to a large extent, and it provides, therefore, a basis for the model calculations discussed in the foregoing sections. For a recent review refer to ZEYHER (1978).

b) The dielectric function method. The basic ideas of this method have been developed in recent years by several people, especially by BAYM (1961); SHAM (1965, 1969); KEATING (1968); SINHA (1969); GLISS and BILZ (1968); and PICK et al. (1970). While these papers focussed on the general structure of the theory and established some important sum rules, the development during the last years has concentrated on the importance of localized (Wannier-type) electronic wave functions for an explicit solution of the problem (HANKE, 1971; PICK, 1971; SHAM, 1972).

Here we shall give a condensed summary of the results of the theory by discussing an explicit form of the dynamical matrix and some important implications (refer to SINHA, 1973, 1978).

The dynamical matrix may be written as

(6.3) 1<"


with

4nZ"Z"e2 L (q + QMq + Q'){3

Vo Q.Q' iq + Q'i 2

X e- 1(q + Q, q + Q') exp[i(Qr" - Q'r",)] (6.4)

with the Fourier transform from ordinary space:

1 • e- 1(K, K') =2 S e- 1(r, r') exp[i( - K r + K' r'] d3 rd3 r'. (6.5)

v

Here K == q + Q denotes an unrestricted wave vector. Q denotes a reciprocal lattice vector, Z e is the bare charge of the nuclei, and e- 1 is the inverse dielectric function. In order to keep the notation from becoming too cumbersome we assume only one atom per elementary cell. This does not restrict the conclusions which follow in this section.

The dielectric function e is well known for the case of a free-electron gas in the Hartree approximation (LINDHART, 1954). In this case e becomes a scalar which depends only on the wave vector q, since the free gas does not experience the discrete ionic structure which in principle leads to Umklapp scattering processes. In many metals a slight generalization which is often sufficient is that the dielectric function is assumed to be diagonal in Q, e=eo(K); this implies that two-body forces are sufficient for a description of the lattice vibrations. The reason is that in a metal the diagonal part of the dielectric function becomes infinite at long wavelengths so that it dominates the effective ionic interactions. This is not true for an insulator where the dielectric function is finite for all wavelengths, due to the electronic energy gap. Nevertheless, even. in this case it sometimes seems useful to separate a diagonal part eo from the whole dielectric function. We may then write the dynamical matrix of a monatomic crystal in the following form (cf. HANKE and BILZ, 1972; ® denotes a direct product):

D(q)= 4nZzez L K@zK+ 4nZ 2 eZ L K@zK(_1 __ 1) Vo Q K Vo Q K eo(K)

ZZ [ v(K) ] [", V(K')] +-L LK(Kfs(K))-(K)@Sss,(q)LK(Kl,(K) (K')

Vo s,s' Q eo Q' eo

where == DCOUL(q) + DDIA(q) + J)MB(q), (6.6)

COUL_ 4nZ2e2 L K~K{3 D~{3 - 1(2'

Vo Q (6.7 a)

DDIA= 4nZ 2 eZ L K~K{3 (_1 __ ) ~{3 K2 (K) 1, Vo Q eo

(6.7b)

JY:1.B- Z2 L t , ~p -- Tt;,(s) Sss,(q) Wp (s), Vo ss'

(6.7 c)

with

Tt;,(s) = ~ K~(Kfs(K)) :o~i) (6.8)


and S(ss') = Sss,(q)· 1. (6.9)

The quantities f.(K) and v(K) are explained below. Here, D;?UL(q) describes the electrostatic coupling between the nuclei. The matrix DDIA(q) describes screened two-body interactions and might be used for the calculation of the rigid ion part. In simple metals, with a scalar screening function 8 0 , the first two terms, D COUL and DDIA, correspond to the free-electron model. Usually, one puts 80 = 1 in insulators which leads to a vanishing DDIA. Then, the matrix IJMB(q) contains, in addition to the two-body electron-ion forces, the many-body part of the lattice. forces and provides the microscopic basis for a generalized dipole model which may be applied to metals as well as to insulators and semiconductors. We can see this by discussing in somewhat more detail the different terms in IJMB(q).

To obtain (6.6) 8 is represented in the Hartree approximation (SINHA, 1973)

e(K, K') = bKK, _ v(K) L f (n, k) - f (n:, k + q) n,n',k E(n, k)-E(n, k+q)

x (n, kl e-iKrln', k+q) (n', k+ql eiK'rl n, k). (6.10)

In this expression f(n, k) is the Fermi function of a Bloch electron with wave vector k, band index n, and energy E(n, k). v(k) is the Fourier transform of the electron-electron interaction including local corrections for exchange and correlation effects (see, for example, HEINE and COHEN, 1970).

With the help of a unitary transformation of the Bloch function In, k) to generalized Wannier function 4>n(r - XL)' the dielectric function can be written:

8(K, K')=80 (K) bKK, - v(K) L As(K) Nss,(q) A;' (K'). (6.11) ss'

8 0 (K) is a conveniently chosen diagonal part of the dielectric matrix. In metals, 8 0(K) may be represented by the expression obtained from (6.10) by putting Q and Q' both equal to zero. The As's are overlap integrals,

(6.12)

with the condensed notation s=(n, n', L). These integrals can (in a dipole approximation) be expressed as three

dimensional scalar products, As(K)=K ·fs(K) , (6.13)

where the physical meaning of the vector fs(K) is that of the K'th Fourier component of a dipole-distribution form factor (cf. HANKE and BILZ, 1972; HANKE, 1978). The polarization matrix N(q) is given by

N ,(q)=L f(n,k)-f(n',k+q) e-i(k+q)(rL-rnl. ss k E(n, k)-E(n', k+q)

(6.14)

The explicit factorization of Il described in (6.11) makes its inversion possible, if N may be approximated by a matrix of relatively small dimensions in the band indices. This leads, finally, to (6.6) with the generalized polarization matrix S(q)

S-1=N- 1+V=N- 1(I+NV) (6.15)


where (6.16)

The general structure of JY1B is that of a dipole-dipole interaction between screened dipole distributions fs/so via a polarization matrix S which contains all localized and charge-transfer electronic excitations involved in the indirect ionion interactions.

The relation of this generalized "screened dipole model" to the usual form of the macroscopic dipole models like the shell model becomes apparent, if we use a representation where the dipole distributions are centered on the ion lattice sites XL" If, for convenience, we define (HANKE, 1971) averaged dipole factors and polarization matrices by:

I f.(K) Nss,(q) fs,(K') == /(K) N(q) /(K), (6.17) ss'

we obtain for the dynamical matrix JY1B

I W(s) S(ss') W(s')* = W8Wt. (6.18) ss'

With this, the following "shell model" equation can be derived (SINHA, 1969, 1971; HANKE and BILZ, 1972 (see (4.18) and (4.29))):

.. t(q) Uion(q)+o-t(q) wshell(q) =0. (6.19)

where the electron-ion matrix .. t is equal to tv, the electron-electron matrix 0-

is equal to 8- 1 and the electronic "shell" coordinate wshell is given by

Whell(q) = _ 8(q) I (Kf(K)) v(K) (K· UiOn(q)), (6.20) Q so(K)

with Uion(q) equal to the polarization vector of the phonon of wave vector q and v(K) the Fourier component of a local pseudopotential.

The microscopic theory in a form as discussed in this section provides a useful basis for further investigations of phonons in very different types of crystals. In particular it may allow the classification of interrelations between the model theories (as reviewed in Sect. 4) and a proper microscopic treatment. We discuss this in the following sections.

c) The direct 'frozen-in' phonon approach. Some time ago, the microscopic theory of phonons in semiconductors was approached by starting from the Hohenberg-Kohn-Sham local-density functional formalism (HOHENBERG and KOHN, 1964; KOHN and SHAM, 1965; SHAM and KOHN, 1966). The main point of this 'direct' approach (CHAD! and MARTIN, 1976; WENDEL and MARTIN, 1978, 1979a, b) is a self-consistent treatment of charge densities and the use of a realistic non-local pseudo potential (YIN and COHEN, 1980). Some principal phonons in silicon could be predicted with less then 2 % errors. Recently, the method was extended to partially ionic crystals, i.e. GaAs (KUNC and MARTIN, 1981). Here, two problems had to be faced: long-range forces leading to effective charges, and low-symmetry phonons with eigenvectors not completely predicted by symmetry.


In their calculation of the self-consistent longitudinal effective charge Z 1 in GaAs MARTIN and KUNC (1981) obtained a value which corresponds to a transverse effective charge ZT= CooZL of l.72[e[ in good agreement with the eX]1erimental value of 2.16[e[ (Table 6.1).

At the present time, one is able to calculate a variety of dynamical properties with this method, such as lattice constant, bulk modulus, phonon frequencies at high-symmetry points (e.g. r and X), Grlineisen parameters and effective charges. Of particular value is the determination of phonon eigenvectors which allows a critical analysis of model eigenvectors. It turns out that some models fitted to experimental dispersion curves give qualitatively wrong eigenvectors (for example, predicting in GaAs the wrong sublattice being at rest in a zone boundary phonon; KUNC and MARTIN, 1981) and should not be used in a calculation of infrared or Raman spectra. The method still requires an interpolation scheme with a (sensible) model for calculation of densities and a high numerical effort to obtain low-symmetry phonons using the HellmannFeynman theorem (KUNC and MARTIN, 1981).

d) Charges and polarizabilities of ions and bonds. In the previous sections we have analysed the lattice dynamics of solids, in particular that of simple diatomic cubic crystals. We have done this in order to provide a basis for the discussion of infrared and Raman spectra in the following sections. Here we are going to summarize the main features of phonons in different types of solids with emphasis on those quantities which are important in light absorption and scattering by phonons. This gives us a starting point for the understanding of these processes in terms of structure, chemical bonding and dynamical interactions in solids.

The treatment of phonons in terms of localized quantities such as point charges, dipoles, polarizabilities, etc. and (except for the Coulomb forces) shortrange nearest and weak second-nearest neighbor forces turned out to be quite successful because of the explicit consideration of electronic degrees of freedom. Formal long-range interionic forces are often due to short-range adiabatic electron-ion forces, and it is this aspect of the theory ·of lattice vibrations which allows a description of phonon dispersion curves with very few parameters, for example, in the case of ionic crystals in a shell model or, in covalent solids, using a bond-charge model. The final justification for these different approaches is given by a successful extension of these models to a treatment of phonon spectra observed in light absorption or scattering.

We concentrate on the simple case of diatomic solids. There exist obvious extensions to the case of more complex crystals. The natural division in the treatment of insulating solids seems to be into ionic and covalent solids, a distinction very familiar to chemists (refer to PAULING, 1940; PHILLIPS, 1969). We have seen in the foregoing discussions that the main difference between these two prototypes is the opposite behavior of the electron-ion interaction. In ionic solids we may consider this interaction first in order to define free (positively or negatively charged) ions; the crystal is then formed by the attractive forces between the ions and is stabilized by the repulsive overlap forces between the valence electrons. In covalent solids the forming of pair


bonds is the decisive point which means that the originally independent atoms in a bond lose their individual character; here the attractive electron-ion interaction overcomes the repulsive ionic forces; this leads to a hybridization of nearly degenerate atomic levels and consequently to an accumulation of electronic charge between the atoms. There exists no gradual shift from one limiting case to the other. This can be seen from the structural transition of tetrahedrally coordinated crystals to those of the rocksalt or cesium-chloride structure, if the ionicity exceeds a certain critical value. In the discussion of phonons this situation was reflected in the use of two different models: the shell model in one case, where the different ionic and electronic multi poles are defined completely at the ionic lattice sites, and the bond-charge model in the other case, where the deformable part of the electronic charge density between the ions is treated as an independent entity often called a bond charge. In short, we may say that in ionic solids (such as the alkali halides) the deformable part of the electronic charge density is localized at the anion lattice sites while in covalent solids this charge density is localized between the ions. The fundamental distinction between 'ions' and 'bonds' is based on this difference.

So far, we have assumed that the third parameter in a solid, namely its 'metallicity' is relatively weak (as, e.g., in LiF or diamond). Metallicity describes the delocalization of the bonding electrons which means that the promotion energy for an electronic charge transfer goes to zero. In such a case, for example in a zero-gap crystal like grey tin or HgTe, both the simple ionic and covalent pictures are inadequate. Since here the screening of charges by the electrons dominates the situation, the most natural description is obtained by assuming a nearly-free electron gas which keeps the positive ions together as in a neutral plasma. It should be noted that this limiting case may be obtained by a generalization of the covalent picture of bonds but not in the ionic case: an extreme ionic polarizability destroys the basic concept of welldefined and separate ions.

The three characteristics of a solid, namely covalency, ionicity and metallicity, are usually of different importance and behavior in different energy regimes. In the static-energy limit where we do not consider any motion of the ions we have a close relation between structure or stability and static ionic charges or bond charges. We may calculate, e.g., the cohesive energy in terms of those quantities. If we consider the crystal in a low-frequency external field as in infrared absorption we have to introduce the corresponding dynamical quantities, i.e. transverse effective charges. etc. Only in the limiting case of weakly polarizable ionic crystals (LiF, etc.) static and dynamic charges are not very different while in strongly covalent systems such as GaAs the static and dynamic ionic charges may differ by an order of magnitude and show no inter-relation whatsoever (refer to HASS, 1967). Finally, in purely covalent materials like diamond infrared absorption is caused by so-called non-linear dipole moments which may be described in a model by using the electronic bond charges as effective charges. This type of infrared absorption is therefore more related to Raman scattering since here the fundamental interaction of the external light is with the electrons of the system while the phonons come into play only by their weak modulation of the electronic polarizability. Again, the description of the


crystal polarizability is quite different in ionic as compared with covalent crystals. The linear polarizability showing up in optic phonons can be described in terms of individual ionic polarizabilities only in strongly ionic crystals where the effective field acting on the ions is close to the Lorentz field. This ionic concept already becomes doubtful in crystals like MgO which still have the 'ionic' rocksalt structure but exhibit a strong delocalization of the outer 2p6

charge density of the oxygen ion. This becomes even more obvious if we analyze the non-linear Raman polarizabilities where the change of the crystal polarizability due to phonon modulation is measured. Contrary to what one might expect at first glance, this change is not a local property of well defined and stable ions (Li+, Na+, Sr++ etc., F-, Cl-, etc.) since here the outer filled electronic shells are rather rigid with respect to a change of their wave-functions. Therefore the easiest way for these ionic solids to change their polarizability is a low-energy charge transfer from the anions to the cations which may be described in terms of interionic Raman polarizabilities, i.e. a real pair or overlap (but not 'bond') property. In covalent solids, on the other hand, the change of the polarizability of a bond is related to the dynamical change of the charge accumulation between the ions, i.e. a change of the hybridization of the wavefunctions. This cannot be achieved in terms of rigid bond charges but more conveniently by defining bond polarizabilities or (because of the high symmetry of the system) equivalent atomic deform abilities with intra-atomic non-linear polarizabilities (refer to Sect. 19). Somewhat surprisingly, this description can be carried through to the oxidic II-VI compounds, even those with sodiumchloride structure (e.g. MgO) where now the 'bond' polarizability is centered around the oxygen ion which dominates the Raman spectra with its intra-ionic non-linear deformability. This seemingly 'covalent' behavior of MgO and similar compounds is connected with the fact that 0- - is not stable as a free ion, which means that the 2p charge density of 0 - - can relax in response to a displacement of neighboring ions. In crystals like SrTi03 , where two of the oxygen neighbors (Ti) are very different from the other four (Sr), an important (linear and non-linear) anisotropy of the oxygen polarizability is observed which is intimately connected with the properties of the ferro-electric soft mode at q = O. We may, therefore, say that in SrTi0 3 and similar crystals the O~ Ti pairs are more covalent than the O~Sr pairs, although the macroscopic dielectric properties are still isotropic due to the cubic symmetry of the crystals. In chainlike systems like Se and Te or other low-symmetry crystals the anisotropy is showing up in the dielectric tensor and consequently in the optic phonons, infrared absorption, etc.

This introductory discussion may have shown how useful the chemical distinction between (predominantly) ionic and (strongly) covalent crystals is for a qualitative survey of the dynamical properties of solids, even if the quantitative application of these concepts to the description of the measured quantities is still restricted. There is, at the present time, a strong and increasingly successful effort going on to establish more quantitative links between the macroscopic, the microscopic and the model quantities and we are going to sketch a few of them with respect to their importance for a physical understanding of phonon spectra.


e) Electric fields and effective charges in ionic solids (FROHLICH, 1949; SZIGETI, 1973). The discussion of infrared absorption and Raman spectra in the following sections requires a knowledge of the dipole moments connected with lattice vibrations. In equilibrium, the net dipole moment is zero in nonferroelectric insulators. Then a dipole moment is obtained which is, in a first approximation, linear in the ionic displacements

M=I mv mL=ZLeuL' L

(6.21)

where Z Leis the ionic effective charge of the L-th particle. Since the interaction of the external field with the crystal is proportional to the induced dipole moment M, the magnitudes of the charges ZLe determine the strength of this interaction.

If the crystals were composed of rigid ions, the bare charge of these ions would be identical to Z L e. In real crystals, the charges of the ions are screened to effective charges during the ionic motion because of the deformation of the electronic charge density. Consequently, the ionic charge has to be described by an effective charge tensor Z', since mL might, in general, not be parallel to uL" This charge tensor contains two parts: the scalar static charge Z, that is, the static equilibrium value, and the polarization charge which describes all the effects of the electronic charge deformation, i.e. Z' = Z + Zp.

In Sect. 4b, Eq. (4.26), it has been shown that the shell model gives a description of the polarization charge in q space of the form zP = - TS- 1 Y, which in alkali halides can be approximated at long wavelengths by Z~ = 0, Z~ = - Ro K= 1 Y_. This means that the polarization charge depends on the product of the short-range overlap forces (given by R o), the dipolar deformability of the negative ions (proportional to K=l), and an electronic effective charge Y_ localized at these ions. zP becomes identical, in the approximation used here, to the mechanical or short-range polarizability d_ of the negative ions (COWLEY et al., 1963).

The ionic effective charge is generally a q-dependent tensor since a correct treatment of the overlap forces leads to q-dependent static charges (Sect. 4c) and since the ionic polarizability is a q-dependent quantity in the partly covalent crystals (Sect. 5b).

We shall summarize here the interrelations of electric fields and effective charges in crystals to provide a basis for the discussion of experimental spectra in the following sections (cf. SZIGETI, 1971; MARADUDIN et aI., 1971). The discussion is restricted to the non-retarded regime where q is still large compared to OJ/c. In this regime the macroscopic Maxwell's equations for an insulating crystal without an external field, i.e. E == Ei , are

div D =div(Ei +4nP)=0, (6.22)

and (6.23)

The magnetic field is neglected in this (electric-dipole) approximation. The internal electric field Ei and the polarization P are assumed to be linearly polarized steady-state macroscopic fields in the crystal. They can be split


uniquely into their irrotational (longitudinally polarized) and solenoidal (transversely polarized) parts. From (6.22) and (6.23) one obtains

E:=O, (even if pt * 0), and

(6.24)

(6.25)

The internal field Ei is therefore zero in a mode with transverse polarization, while in a longitudinal mode it is proportional to the polarization but of opposite direction. In view of (6.25) the displacements u of the lattice particles uniquely determine the internal field E:, which forms part of the restoring force.

If, in addition, an external (radiation and Coulomb) field is present, the total field in the crystal is a sum of the transverse external and longitudinal internal and external fields:

(6.26)

The total field is assumed to be a macroscopic field with a wavelength much longer than the atomic dimensions. Then, we still have div D = 0, but now rot E = rot Ee * ° so that the total field exhibits a transverse" external" field while the transverse "internal" field (that is, that part of the field in the crystal due to internal sources) is still zero.

In order to obtain a relation between the macroscopic field E and the polarizability of charged particles in the crystal, we introduce the effective field Eeff which is defined as the field acting on the particle excluding the dipolar field of the particle considered. The dipole moment of the particle is then

(6.27)

where oc is the polarizability of the particle. The effective field can be calculated, following Lorentz, by introducing a

sphere around the particle which is assumed to be small compared with the wavelength of the electric field inside the sphere. The field due to charges inside the sphere is called the Lorentz field which depends on the polarization as

4n EL=-3P=E-Em. (6.27a)

The total field E is the sum of the Lorentz field EL and the field Em originating from all sources outside the sphere in the surrounding medium. The field Em and the effective field Eeff,m arising from the surrounding medium are identical for a sufficiently large sphere. On the other hand, the effective field inside the sphere, E eff, s' is due to all charges inside the sphere except that of the particle considered. It is quite different from EL and generally a complicated quantity.

The effective field acting on the particle is then

Eeff=Eeff,m + Eeff,s

=E-EL +Eeff,s 4n

=E+3 P+Eeff,s. (6.28)


For electrically isotropic structures with point dipoles (that is crystals where each ion lattice is of tetrahedral or higher symmetry) it can be shown that Eeff , s = O. Then,

(6.28 a)

Equation (6.28a) may be a good approximation for ionic crystals such as alkali halides where the constituents of the crystal are nearly spherical, non-overlapping ions.

In a strongly ionic crystal with weakly overlapping ions, such as an alkali halide, the polarizability rx of the crystal can be expressed approximately as the sum over all individual contributions of negative and positive ions, so that the polarization is given by

p= :0 (rx++rx_) (E+ 43n p). (6.29)

Since the polarization P is generally related to the macroscopic field E by

4nP=(s-1) E, (6.30)

the Clausius-Mossotti relation

(6.31)

is obtained, where sand ()(+, _ are frequency-dependent quantities. Equation (6.31) can be extended to crystal structures where Eeff,s=l=O. This means that the Lorentz field correction in (6.29) has to be modified to a more general quantity. In addition, the overlap between neighboring ions in an ionic crystal should lead to a generalization of (6.31) where the sum of the independent polarizabilities of negative and positive ions is replaced by a more general molecular polarizability r:I. which could be derived, in a first approximation, from the quantum mechanical expression (BA YM, 1969)

r:I.=2 L 1<01 M Ii) <il M 10). i Ei-EO

(6.32)

Here, Eo is the energy of the crystal ground state 10) and Ei that of an excited state Ii). If 10) and Ii) can be represented approximately by (symmetrized) products of the wave functions of negative and positive ions,

(6.33)

(6.31) must be generalized to an equation, where "charge-transfer" terms are added to the individual polarizabilities. Such a treatment might, in principle, be expected to overcome some of the difficulties in the lattice dynamics of multiatomic and partly covalent crystals (cf. SINHA, 1973).

We note that in practice the longitudinal effective field acting on a nearly free electron does not differ from the macroscopic field; thus the Lorentz field correction is zero in this case. This result is plausibJe since the local correction


to the total field by a very spread-out charge distribution is negligibly small; therefore a nearly free electron always experiences the macroscopic field E.

Since localized point charges and homogeneously distributed free electrons describe different limiting cases of the charge distribution in crystals, the local field correction to the macroscopic field might be described by the general formula (GUERTIN and STERN, 1964)

4n Eeff=E +3 yP, (6.34)

where the local-field tensor y is a delocalization and anisotropy factor which equals unity for the cubic point-ion lattices (as in alkali halides) and zero for the free-electron gas (as in metals), but could be a complicated tensor in intermediate cases which is determined by the symmetry of the crystal but does not depend upon its shape (DE WETTE and SCHACHER, 1965). The ClausiusMossotti relation (6.31) then is changed to

£-1 4n rx

3+y(£-1) 3 Vo

which gives in the free electron limit (y = 0),

£=1+4nrx/vo·

(6.35)

(6.36)

Usually, one expects y to be almost unity even in cases where an appreciable amount of the electronic charge density at a certain ion lattice site is delocalized (GUERTIN and STERN, 1964; see also: THEIMER and PAUL, 1965; STERN, 1966). For recent discussions of the microscopic theory of local-field corrections see HANKE (1979) and BONNEVILLE (1980). From Bonneville's treatment it can be seen that the local-field factor y in cubic crystals is approximately given by 1 - f, where f is the "charge overlap", i.e. the fraction of ionic charge that spreads outside the primitive cell. For monatomic crystals such as the alkali and the noble metals y is about 0.3 which corresponds to a charge overlap of f c.::::0.75. In diatomic covalent crystals (Si, Ge) the average value of y is about 0.50 while in alkali halides it is much closer to one. Here f measures the overlap between two sites of the same sublattice i.e. between second-nearest neighbors which is small in alkali halides.

We may write the macroscopic polarization of the crystal related to lattice vibrations approximately as a linear expansion with respect to two independent variables, namely the external field Ee and the nuclear displacements u which represent the internal field Ei

p= - E + - u (ap) (ap) aEe u e au Ee

(6.37)

= [(:;t + G!te' :;J Ee' (6.38)

In going from (6.37) to (6.38) we have used the fact that we are considering the case where the ionic displacements are in dynamical equilibrium with the external field. The first term in (6.38) is the purely electronic susceptibility Xel'


while the second gives the vibrational part Xvib' which includes the adiabatic

deformation of the electronic charge density. V ~P is the effective charge tensor of the lattice vibrations. U

Considering a long-wavelength optic wave in an alkali halide, we have the potential energy per ion pair

(6.39)

with Jl the reduced mass, U the relative displacement of the ions, and j the polarization (longitudinal or transverse). The dipole moment connected with u is proportional to u in the linear approximation, so that the polarization is

. 1 PJ=-Zjeu,

Vo (6.40)

where Zj e defines the effective charge for the vibration under consideration. As discussed above, P (and therefore u) determines uniquely the internal field Ei for long-wavelength longitudinal vibrations, so that in this limit the longitudinal effective charge ZLe and the frequency WLO are practically independent of the wave vector. This quasi-static behavior holds also for the long-wavelength transverse vibrations due to the homogeneous polarization of a large crystal volume and the neglect of retardation.

From (6.38) we obtain the macroscopic susceptibility

(6.41)

where j denotes the transverse or longitudinal polarization. Assuming one linearly-polarized external wave, the potential energy per ion pair is then

(6.42)

For small frequencies of the external field, W~Wj' the force due to the displacement u is in quasi-static equilibrium with Ee so that V takes its minimum value with 8Vj8u=0. We introduce the equilibrium value of u into (6.40) for the vibrational polarization and obtain

. Z~ e2

X~ib=~-· Jl W Vo

(6.43)

With (6.24)-(6.26) and (6.30) we can eatablish the following relations between the susceptibilities and the dielectric constant:

and 4nxT=e-1

8-1 4nxL=--.

8

(6.44)

(6.45)

At low frequencies 8 approaches the static dielectric constant 8o, while at high frequencies 8 becomes the electronic value 8 00 since the lattice vibrations can no longer follow the external field. For the vibrational contribution alone, the result is then:

(6.46)


and 8 -1 8 -1 1 1 4nXL. =_0 ____ 00 __ = ___ .

vlb 8 0 8 00 8 00 80

With the help of (6.43), this gives for the effective charges

4n zi e21vo = J1, Wio(80 - 8 00 )

and 22 2(11) 4nnZL e Ivo=J1, W LO --- .

8 00 80

Sect. 6

(6.47)

(6.48)

(6.49)

Equation (6.48) describes the connection between the transverse effective charge and the oscillator strength of infrared absorption, which is important for the discussions in Sects. 15 f£

We may obtain the relation between both effective charges by comparing longitudinal and transverse optic waves in the absence of an external field. From (6.40)

(6.50)

and

(6.51)

Since E~ = - 4n pL and ET = 0 and since the adiabatic electronic response function 8=8 00 , we obtain for the difference of both polarizations

or (6.52)

The longitudinal effective charge is therefore equal to the transverse one in rigid ionic crystals, but is smaller in polarizable crystals by a factor 8 00 due to the electronic screening of the longitudinal internal field E~ Equations (6.48) and (6.52) give the Lyddane-Sachs-Teller relation (4.7).

For the effective fields, we obtain from (6.28) with Ee = 0,

T 4n Eeff=}P,

L 8n Eeff = -}P.

Here, we note for comparison with the literature a specific case:

(6.53)

(6.54)

(6.55)

where E!ff means the effective field for a uniformly polarized sphere for which,

from (6.28), E~ = - 43n P, that is, the Lorentz field. For this case, the correspond-


ing values for the vibrational polarizability lJ(~ib' the effective Szigeti charge Zs e, and w;, may also be calculated:

and

4nlJ(S =3~-3~= 4n Z; e2

vlb 2 2 3 2 , 8 0 + 8 00 + J-l Ws Vo

Z =Z800 +2 T s 3 '

2 8 00 +2 2 WTO=--Ws'

80 +2

(6.56)

(6.57)

(6.58)

In a diatomic ionic crystal, we may use approximately the electronic polarizabilities of the separate ions, so that with (6.31) and (6.35)

(6.59)

With respect to the Clausius-Mossotti limiting case of point dipoles, y = 1, (6.59) leads to a decrease of the susceptibility Xoo ' if y becomes smaller than unity.

We can now see how the three different effective charges are related to the splitting LI w2 == wlo - wio of the optical frequencies:

A 2 4n 1 Z2 2 J-lL.JW =-- T e

Vo 800

4n =-Z Z e2

Vo T L

_ 4n (8 00 +2)2 Z2 2 -~ 9 s e .

(6.60)

(6.60a)

(6.60b)

The transverse charge ZTe is the natural charge to use in a discussion of infrared absorption. The longitudinal charge ZLe was used by Callen in his theory of the electron-phonon coupling in semiconductors (1949). Zs e is the Szigeti charge as discussed in Sect. 4b; this was introduced by SZIGETI (1949) in order to investigate the effect of the electronic-charge deformation on the lattice vibrations in ionic crystals. In fact, Zs should be equal to the static charge Z = ± 1 in alkali halides if electronic polarization were to be neglected, since it is the effective charge of the free vibration of a sphere with uniform polarization and Beff equal to zero, so that no electronic deformation should be involved. This is not true in real crystals. Using experimental values for J-l, Llw2 and 8 00 , it turns out that Zs is close to 0.75e in alkali halides (SZIGETI, 1949). The assumption leading to this discrepancy was made in (6.31) when non-overlapping spherical ions were assumed. The overlap leads to short-range non-electrostatic forces which in addition to the effective field contribute to the electronic charge deformation. In our discussion of the shell model in Sect. 4b the short-range contribution to the effective Szigeti charge was found to be the "polarization" charge ZP. The transverse charge ZT differs from Born's static


charge Z, (4.1), or the Lundquist-Zeyher overlap charge Z', (6.1), by 10-30%. Usually one chooses Z = 1 in alkali halides, i.e. the free-ion value (consistent with an ionic Heitler-London approach). For most of the alkali halides we then find

Zs<Z'<Z<ZT

or, numerically, 0.75 < 0.9 < 1.0 < 1.1.

(6.61)

We recall that the best-fit input static charge for shell models gives values of Z between 0.85-0.95 for alkali halides (SCHRODER, 1972). This indicates that an ad-hoc overlap correction (Z --+ Z') is required for models which do not explicitely consider overlap effects.

This seems also to hold for a recent discussion of the transverse effective charge in ionic compounds by HARRISON (1980). In analogy to his treatment of mainly covalent systems (see subsection 6f) he calculates the effect of overlap polarization on this charge by first-order perturbation theory which leads to the formula

(6.62)

where v'pa is the main interatomic p(halogen)--+s(alkali ion) matrix element, while Eg denotes the corresponding optic p - s band gap. In the simplest form, HARRISON obtains ZT~ 1.16 for all rocksalt-type alkali halides, a value remarkably close to an averaged value of these compounds. Again, replacing Z by Z' in (6.62) may be a more consistent approach because a static (overlap) correction of Z should be considered before a dynamic (polarization) correction is taken into account.

For an illuminating discussion of effective charges the reader is refered to MARADUDIN (1974a, p.62). Of interest is also a recent calculation by VOGL (1978) who used pseudopotential theory for the determination of WT'

In Table 6.1 effective charges for several binary compounds are listed. Note that the Szigeti charge exhibits rather similar values between 0.5 and 1.1 while the transverse charge show values between 0.8 and 9.

The effect of the short-range distortion of the electronic charge density on the effective charges, the bulk modulus, etc. was reviewed by DICK (1965) and COCHRAN (1971) in the framework of the shell model. The change of the highfrequency dielectric constant in the simple shell model is

, (Y Y)2 Goo -1 4n 4n 4n K: - K~ G + 2 =31) IXoo =31) (IX+ + IX_) -3 1 1 1 (6.63)

00 0 0 _+_+_

4n Ro K+ K ~-3 (IX + +IX_), if IX+ ~IX_, (6.64)

Vo

~-IX 1 __ 0 4n ( R') 3vo - K_'

(6.65)

Here the notation of Sect.4b has been used with IX±/vo=Y;K±l. We see that for nearly equally polarizable ions the short-range effects cancel, while for different ions an appreciable decrease of IXoo (and therefore Goo) appears.


Table 6.1. Effective charges of binary compounds

Compound Z, Z' ZT Compound Z, Z' ZT

LiF 0.65 0.95 0.85 CuCI 0.44 1.12 LiBr 0.74 0.95 1.28 CuI 0.96 (1.0) 2.40 KF 0.91 1.0 1.17 AgI 0.61 (1.0) 1.40 KI 0.75 1.0 1.17 BN 1.14 (2.0) 2.47 RbF 0.98 (1.0) 1.28 AIN 1.21 2.75 MgO 1.42 1.85 2.34 GaN 1.3 3.2 SrO 1.22 (1.8) 2.14 AlAs 0.62 2.3 PbS 0.75 (2.0) 4.8 GaAs 0.50 (2.0) 2.16 SnTe (300K) 0.52 (2.0) 8.1 GaSb 0.39 2.15 AgBr 0.67 (1.0) 1.48 InAs 0.53 2.53 BeO 1.10 (2.0) 1.83 SiC 0.89 2.57 CdS 1.15 (2.0) 2.77 TICI 0.86 (1.0) 1.94 ZnTe 0.65 (2.0) 2.00

Z, = Szigeti charge, Z' = static charge (,valency'), ZT= experimental values of transverse effective charge. Definitions are given in the text. Values of

Z" Z', ZT refer to average temperatures between 0 K and room temperature.

For the purpose of the following sections, we may note that the general formula for the dielectric function can easily be obtained from the foregoing discussion by considering a general frequency W in (6.51) and using the dynamical matrix of Sect. 4b (BORN and HUANG, 1954). This yields for the cubic diatomic crystals:

(6.66)

In crystals with more than one dispersion oscillator but still with cubic symmetry (such as the perovskites), (6.66) has to be generalized so that it includes several transverse effective charges corresponding to the number of active oscillators. In crystals with lower than cubic symmetry, Il exhibits different values for electromagnetic waves travelling in different symmetry directions. These generalizations will be discussed in close connection with the analysis of experimental spectra. Here, we give the general formula:

(6.67)

where wL,ai and WT,o:i are the frequencies of longitudinal and transverse optical phonons with polarization vectors parallel to one of the principal axes IX in the crystal.


For a generalization of the Lyddane-Sachs-Teller relation to systems with Debye relaxation (orientational dipole polarization) see BARKER (1975) and CHANG (1976).

f) Fields and charges in covalent solids. As discussed above, in monatomic covalent solids an effective ionic charge can only be observed in infrared absorption, if the symmetry is lower than tetrahedral symmetry, for example in Se or Te. There is still infrared absorption via electronic charges in highsymmetry crystals, and we shall see that the bond-charge model provides a satisfactory description of this electronic effective charge in infrared absorption (Sect. 16). We shall expect that in this case too the effective charge experiences a sort of local field analogous to the Lorentz fields for ionic solids (HANKE and SHAM, 1974; CHADI and COHEN, 1975). The excited states in covalent solids may be described to a good approximation by nearly free electron states (OPW, etc.), which could suggest that only the macroscopic field plays a role in optical transitions (BURSTEIN, 1963). However, since the valence band states in C, Si, etc. are well localized between the ions, a certain local field must influence all transitions, charges, polarizabilities etc. The possible neglect of a local field is related to the condition of sufficient delocalization of the electronic charge density so that every electron only "sees" a mean or macroscopic field. We may conclude that in ionic and covalent solids the local field loses its significance with increasing polarizability of the ions or "metallicity" of the crystals as described, for example, by the dielectric constant eoo. We have seen (Sect. 5e) that the bond charge takes up an asymmetric equilibrium position, if the two ions in a bond become unequal. This asymmetry leads to the sudden appearance of effective ionic charges in the crystal which are related to the asymmetry of the bond-charge position but cannot be obtained by a reasonable division of the electronic charge around the ion cores. - The same difficulty arises with the crystal polarizability where the reduction to bond polarizabilities seems to be the most extreme for a localized description of the dielectric properties (Sects. 18 and 19). - In the last decade several attempts have been undertaken to derive the electronic properties of a crystal from a molecular picture which should be simple enough to describe the trends in large families of crystals by only very few parameters related to covalency, ionicity and metallicity of the crystal in an obvious fashion. We outline two of these "chemical" concepts which have attracted rather general attention because of their conceptual simplicity.

The first to be mentioned is PHILLIPS' (1969, 1970) dielectric oscillator model for the AN B 8 - N family of crystals. He was stimulated by the success of Pauling's concept of electro negativity (refer to PAULING, 1940) and related quantities for a classification and semi-quantitative description of many "bond" properties in diatomic systems. He started from the observation that the energy of the largest absorption peak in e"(w), the imaginary part of the dielectric constant, shows certain clear trends in isoelectronic families, such as the tetrahedral AN B8 - N

compounds. Penn's (1962) formula relates the static value of e to the average band gap Eg h 2 2

e ::=e'(O)=I+a.N ( wp) ::=1+ wp (6.68) 00 2E w 2 ' g g

Sect. 6

with

Microscopic theory, models, and macroscopic quantities

a=1- Eg <1 4E

71

(6.69)

in Penn's parabolic band approximation. This is like the static value of B for an oscillator with the plasma frequency wp and a single transition frequency Wg'

The average gap may be separated into a covalent and an ionic part,

(6.70)

The ionicity is then defined as

(6.71)

It turns out that the A N B8 - N crystals with tetrahedral coordination possess an ionicity smaller than the critical ionicity Ie = 0.785 (PHILLIPS, 1970), while those with octahedral structure show larger h values. The precise value of Ie depends on a in Penn's equation (BRUST, 1971; ALAVASHI et aI., 1972) but this does not influence the qualitative success of the model. The weakest point from a theoretical point of view is perhaps that the homopolar part of the energy gap, Eh , is determined from an empirical relation to the lattice constant,

(6.72) (VAN VECHTEN, 1969).

On the other hand, Eh and C may be considered as being derived from the even and the odd parts of the pseudopotential, starting from the center of the bond. Since the lattice constant does not change very much in an isoelectronic series, Eh is practically a constant which may be. determined from empirical values of Boo and E g , thus giving the values of h. Also here, explicit expressions for Eh and C in terms of second -order perturbation theory of pseudo potentials (HEINE and COHEN, 1969) exhibit certain difficulties with respect to the dipolar sum rule and larger values of the ionic gap C (CARDONA, 1971). This is not surprising in view of the simplicity of the theory and does not affect seriously its usefulness in establishing chemical trends of properties as a function of the ionicity h.

We may ask, how h is related to the charges and forces important in lattice dynamics and infrared absorption. First, we should note that another critical value of h' he = 0.9, separates the" real" ionic crystals such as the alkali halides with a positive pressure derivative of Boo i.e. dBoo/dp > 0 (which one expects classically) from the "pseudo"-ionic ones such as MgO with a negative pressure derivative of Boo' VAN VECHTEN (1969) has correlated this fact with the closedshell behavior in alkali halides as compared to the open-shell situation in MgO and other compounds with h < 0.9, i.e., the onset of hybridization and covalency in these crystals which leads to a rapid increase of the covalent gap Eh • We shall see that this distinction is reflected in the Raman spectra of the isoelectronic A N B8 - N crystals (Sect. 19) but with a modified interpretation.

Another interesting relation is the linear decrease of the ratio of bondbending forces to bond-stretching forces with increasing ionicity h (R.M. MARTIN, 1970). This gives a very simple description of the covalent forces and may,


in fact, be related to the pseudo potential form factor which has a directional part (PHILLIPS, 1972). It is now clear that Weber's bond-charge model (1975) was so successful because it used the corresponding Keating (1968) potential for the description of bond-bending and bond-stretching forces.

The bond-charge in Phillips' model may be identified with that part of the charge density centered between two neighboring ions. It can be assumed to be approximately equal to - 2e in close analogy to the old chemical description of a valence bond. In lattice dynamics there is a screening of the interaction of the different charges which leads to a reduction by a factor of C;~2 thus giving" effective" bond charges of about -0.2e. In this respect the model is similar to an effective rigid-ion model which means that it gives the dispersion curves correctly but does not describe the main part of the crystal pol ariz ability (WEBER, 1975).

The bond-charge model has been extended by LEVINE (1970, 1972) to the case of non-linear optical susceptibilities due to three-phonon non-linear interactions ('optical mixing'). He considered a cubic potential for an anharmonic oscillator with its harmonic part taken from the dielectric oscillator model. He obtained MILLER's (1964) rule about the interrelation between linear and nonlinear susceptibilities. It might well be that an analogous treatment of first- and second-order Raman polarizabilities would be possible, if the bond charge itself would be treated as a deformable or polarizable quantity.

At this point it seems that this successful "chemical" approach asks for a more basic, quantum-mechanical foundation. The three important features in the treatment of solids, namely covalency, ionicity and finally "metallicity" or delocalization (which in Phillips' model is related to the minimum and not to the average gap, refer to PHILLIPS, 1972) should be expressible in terms of matrix elements taken with conveniently defined wavefunctions.

A step in this direction has been taken by HARRISON (1973, 1974, 1975) and by LAUNOO and DECARPIGUY (1973). They express the matrix elements of the Hamiltonian of a diamond-type crystal in terms of atomic Sp3 hybrid orbitals thus obtaining for every bond two intra-ionic matrix elements and one interionic or "resonant" one. The basic idea stems again from chemists (HALL, 1952; COULSON et aI., 1962) and an early attempt by LEMAN and FRIEDEL (1962). The appealing point of this approach is that it explicitly uses the local properties of the "bond" wave functions in ordinary space while the strength of Phillips' approach is its relation to pseudopotential theory and dielectric response theory in k-space. Since both treatments are defined in terms of effective two-level systems it is not astonishing that all parameters of the spectroscopic model of chemical bonds can be more or less identified with the corresponding parameters of the bond orbital theory. One of the most interesting results is that the transverse effective charge ZT e is given in this theory by (cf. LAUNOO and DECARPIGUY, 1973; N = number of valence electrons)

(6.73)

where I is the ionicity or polarity in the bond orbital description:

(6.74)


where A is the asymmetry factor in the bonding or anti-bonding orbital

1 t/lb = (1 +A2)1/2(IPA +AIPB) (6.75)

1 t/lab = (1 + A 2)1/2 (A IP A - IPB) (6.76)

with A = 1 in IV-IV crystals with A = B. We note that Harrison uses the same definition for his 'polarity', ocp, but assumes a value of 2 for e in (6.73). This leads to

ZT = Z* + (8/3) ocp(l-oc~) (6.77 a)

with the 'effective atomic charge'

Z*=4-N -4ocp • (6.77 b)

It turns out that j2 or oc~ are not very different from Phillips' ionicity /;. The crucial parameter in (6.73) is e which is given by

() = (} In(p/R)/(}(P/R), (6.78)

where P is the resonant integral and R the bond length. There are general arguments from pseudopotential theory (BENNETT and MARADUDIN, 1972) that e should not be far from 2. In fact, this parameter shows close analogy with the exponent of Eh '" a- s in Phillips' theory (S '" 2.5). A further but small correction is required for piezoelectric crystals (see HARRISON, 1980). The calculated values of ZT for many II-VI and III-V compounds are in remarkably good agreement with experimental data as obtained from (6.48) (LANNOO and DECARPINGUY, 1973; HARRISON, 1973, 1974, 1980), i.e. between 1.8 and 2.7.

A serious problem in the dielectric model as well as the bond-orbital model seems to be the insufficient treatment of the dehybridization or metallicity which shows up most clearly, if we look into trends within one AN BN - 8 crystal family with constant N but increasing main quantum number. We remember that in Phillips' approach metallicity was related to the minimum energy gap quite similar to Harrison's treatment where the broadening of the bond orbitals into a band by metallicity is associated with a decreasing energy gap. In Phillips' approach the difference between the minimum and the average energy gap contains the information about the dehybridization in a crystal family with a fixed value of N. In Harrison's model we find it difficult to differentiate between the effects of the decrease of the mean gap and the increase of the splitting of the one-particle levels Es and Ep. Recently, HARRISON has given an improved definition of metallicity (HARRISON, 1980).

We shall see in Sect. 18 that the dehybridization of the covalent bonds in the sequence C, Si and Ge is clearly reflected in the parallel increase of the transverse bond polarizability which is equal to zero in the case of diamond only. The simple bonding-antibonding scheme is insufficient in this case, and new quantities such as bonding-antibonding mixing or, in a tight-binding scheme, n-bonds have to be considered (refer to HANKE and SHAM, 1975). This


gradual transItIon to an increasingly metal-like charge distribution is also accompanied by an increase of the bond length. If this length exceeds a certain value, for example due to melting (SZIGETI, 1976), covalent compounds often become metallic.

g) The microscopic description of charges and fields. In Sects. 6a and 6b we have described the microscopic theory of phonons in terms of the overlap theory and the dielectric function approach. Here, we summarize the most important results of that theory with respect to charges and fields after having illuminated the basic physical concepts in the foregoing section.

The overlap theory starts from the concept of rigid ions with filled shells and uses the overlap of wavefunctions mainly between neighboring ions as an expansion parameter. Obviously, the theory is very appropriate in the case of strongly ionic crystals with little covalency or metallicity. The prototype of this class of crystals are the alkali-halides with lower ground-state main quantum numbers, where a simple shell or deformation-dipole model is sufficient. Recently ZEYHER has shown (1975) (see also the comment by JASWAL, 1975) how to identify the model parameters with certain microscopic quantities. The ionic polarizabilities and deformabilities (Sect. 4d) may be brought into correspondence with Hartree-Fock polarizabilities (DALGARNO, 1962). The shell charge Y or, more precisely, its ratio to the n.n. repulsive force constant ;;::;A is given by the ratio of the long-range electronic polarizability !xoo divided by another polarizability fi. which contains both a long-range and a short-range part:

(6.79)

SCHRODER (1971) has found that Y_ in the alkali halides is nearly a constant quantity of about - 2.5. In (6.79) this means that fi. may be approximately factorized by taking out of it an averaged short-range deformation potential cx::A. Such a proportiomility oflong-range and short-range forces is known from the equilibrium condition. In (6.79) it would mean that the deformation potential may be approximated by simple analytic potentials of a given power (TOLPYGO, 1961). The polarizability fi. is, except for a numerical factor, equal to the polarization charge ZP, i.e. the difference of the Szigeti effective charge Z' and the static charge Z, Eq. (4.26). These interesting relations show that the overlap theory should be an appropriate starting point for the investigation of non-linear phenomena such as infrared absorption and Raman scattering which are intimately conneeted with the effective charges and the ionic deformabilities in ionic crystals.

The dielectric-function approach came historically from the dielectric theory of simple metals, where ionicity and covalency may be completely neglected. This is related to the fact that here the dielectric constant at long wavelengths is approximately given by its static limit, 8( co = 0, q) -1 cx::q - 2 where it dominates all other effects. The interesting deviations arise, if we leave the regime of pure metals and extend the theory to the case of covalent or ionic insulators where the static limit of the electronic dielectric constant, 8 00 , is finite. We have seen in


Sect. 6b that the different parts of the dynamical matrix refer in a natural way to a metallic part (given by some diagonal Bd), a rigid ion part (B~ 1) and the rest of the dynamical forces (i.e. screened dipolar forces), containing covalency and polarization effects. Of particular importance is the off-diagonal part of the dielectric matrix, B(K, K') with K =j=K', since it contains all types of local field effects, i.e. those parts of the electron-ion interaction which depend on the discrete nature of the lattice. The relation of B, in particular of its off-diagonal elements, to various macroscopic quantities in ionic and covalent solids has been reviewed in several recent articles (SINHA, 1973; SHAM, 1974; BILZ et aI., 1974; SINHA et aI., 1973; PRICE et aI., 1973).

We note first that the optical dielectric constant is given by

Boo = lim {B-1(q,q)}-1 q~O

=lim{B(q,q)- L B-1(q,q+Q)B(q+Q,q+Q')B(q+Q',q)}-1 (6.80) q~O Q,Q'*O

where the second term in the curly brackets denotes the local field corrections to Boo' In the point-dipole approximation for ionic crystals this leads to the Lorentz-Lorenz formula, since then (SINHA et aI., 1974)

a B(O, 0) = 1 + 4n-

Vo

where a/vo means a cell polarizability, and

4n (a)2 -1 4 a TY ~ 1 4

Boo - + n-+ 4 + nxoo'

Vo 1-~y~ 3 Vo

(6.81)

(6.82)

Here we have added an Adler-Wiser factor y, 0 ~ y ~ 1, which takes account of the delocalization of the electronic charge (cf. (6.35)). We may use (6.82) as an interpolation formula for going from the strongly ionic case, y ~ 1, to the weakly ionic case, y ~ 1. For the situation in covalent tetrahedral crystals such as silicon, see VAN VECHTEN and MARTIN, 1972.

The transverse effective charge ZT is obtained from the bare ionic charges ZK in cubic crystals

(6.83)

with the screening charge (BILZ et aI., 1974)

zsc Z I' '\', - l( Q) q . (q + Q) [' Q ] K =Boo K' 1m L., B q,q+ Q)2 exp 1 rK •

q~O QH (q+ (6.84)

In order to ensure charge neutrality and, consequently, the existence of stable acoustic modes, the sum of the effective charges has to vanish

(6.85) K K

This acoustic sum rule may be derived from the infinitesimal translational invariance of the system (SHAM, 1969; PICK et aI., 1970; KEATING, 1968, 1969).


Its simplest representation is in the rigid-ion model, for example in alkali halides. If we treat the 6 outer p-electrons of the anion as contributing to e, thus starting with the bare charges Z 1 = + 1 and Z 2 = + 5, we have the screening charges Z~c=O and Z~c= -6 so that (6.85) is fulfilled in a trivial way with the rigid-ion charges 21 = Z 1 = + 1 and 22 = Z 2 - Z~c = - 1. It can easily be shown (BILZ et aI., 1974) that the replacement of the bare charges corresponds to a charge renormalization with the help of the off-diagonal elements of Il thus leading to a new dynamical matrix with rigid-ion charges interacting via nearly 'diagonal' screened dipoles and a small polarization part left:

e- 1(q, q + Q)=e;;; I eR/(q, q +Q)+epot(q, q +Q),

or, in terms of the polarization charge introduced earlier

with -1

Z~OL= -lim eo(q) ZK If epOL(q, q +Q). q~O Q

(6.86)

(6.87)

(6.88)

Since the rigid-ion charges 2" fulfil by themselves the acoustic sum rule the remaining off-diagonal part BroIL of B may become small and might be calculated in a perturbation scheme. The 'best choice' of the rigid-ion charge is subject to overlap corrections as has been discussed above. If one tries to minimize ZPOL at w =0 one can introduce a screened rigid-ion charge

(6.89)

As a consequence the Lyddane-Sachs-Teller splitting (6.60) is automatically fulfilled. Although debatable from a microscopic point of view (no polarizability is obtained, cohesive energies cannot be calculated), it may be a useful model for complex ionic crystals, where an explicit consideration of polarization effects is impossible.

The analogous treatment of B in the case of covalent solids has been discussed in detail by SINHA et aI. (1973). Here, one has to consider the charge accumulation between the ionic charges. If we start from the bare ionic charges ZK(~ +4) at the ion sites, we have to introduce screening bond charges Zb= - Z,,/2 between the ions in order to ensure charge neutrality in a cell

(6.90)

Zb may be calculated from (6.84) by centering it at the midpoint of the bond (BERTOLANI et aI., 1973; TOSATTI et aI., 1972). In analogy to (6.89) the 'best choice' of Zb is obtained by using the effective bond charge

(6.91)

The similarity of the bond charge model (PHILLIPS, 1969; MARTIN, 1970; WEBER, 1974) with an effective rigid-ion model thus becomes obvious. A discussion of polarization effects would require the introduction of a polarization charge Z~OL as given by (6.88).

Sect. 7 Theory of interaction of photons with particles 77

The foregoing discussion shows the usefulness of the microscopic theory of phonons for an understanding of the physical meaning of model parameters and macroscopic quantities. We note finally that a unified treatment of phonons in crystals including covalent, ionic and metallic forces can be obtained in the model theory (refer to WEBER et aI., 1972) as well as in the microscopic treatment (see HANKE and BILZ, 1972).

c. Interaction of photons with matter

7. Theory of interaction of photons with particles. In this chapter the general features of the theory of infrared absorption and of Raman scattering will be described. We are interested in processes where the absorption or scattering of light results finally in excitations of phonons in the crystals. There generally exists a marked distinction between the fundamental processes in light scattering and infrared absorption. Infrared absorption involves mainly the anharmonic interactions between phonons which are intimately connected with lifetime of phonons, etc. Raman scattering, on the other hand, is due to electron-phonon interaction related to fluctuations of the crystal polarizability.

We note that infrared absorption in covalent crystals such as diamond involves the electronic charges in the induced electronic dipole moments. These dipole transitions are directly seen in the electronic optical absorption but their details, which reflect the electronic band structure, are only important in the resonant Raman effect which we shall briefly sketch in Sect. 9 g.

In the main part of our discussion electronic excitations play only a role as virtual intermediate states. In Chap. B it was described how they dress the phonons by leading to polarizabilities, deformabilities, effective ionic charges and bond charges. The excitation energies for real electronic transitions in nonmetals are high in comparison with phonon energies. Here, in infrared absorption and non-resonant Raman scattering, the exciting photon energies are still low so that we are able to examine the above-mentioned concepts for adiabatic electrons when going to non-linear (phonon-phonon and electronphonon) processes.

We note that in light of the microscopic theory (Sect. 6) the different role of the electrons in infrared and Raman experiments is closely related to the difference between off-diagonal and diagonal parts of the dielectric function B.

This aspect makes contact with the discussion of effective charges, etc., in Sect. 6b.

a) Non-relativistic theory of inelastic scattering

IX) Some basic quantities. The theory of scattering of photons by phonons has been described in detail in many books. The reader is refered to the recent presentations of the theory of Raman and the related Brillouin scattering by

78 Interaction of photons with matter Sect. 7

POULET and MATHIEU (1970, 1974), ABELES (1972), ANDERSON (1971, 1973), BIRMAN (1974), CARDONA (1975), and HAYES and LOUDON (1978). The books by POULET and MATHIEU, ABELES and BIRMAN also contain presentations of the theory of infrared absorption. The related theory of neutron scattering due to phonons may be found by the interested reader in the books by MARSHALL and LOVESEY (1971) and LOVESEY (1977).

Therefore, it seems to be sufficient for the purpose of this article to summarize general concepts and important results thus providing a basis for a systematic discussion of phonon spectra. A few points will be discussed in some more detail, for example the relationship between gauge invariance and the explicit frequency dependence of cross sections and also the different sum rules which govern infrared absorption and Raman scattering.

Let us, first, define some basic quantities of light scattering. For this purpose a typical light scattering experiment is schematically shown in Fig. 7.1. Here, we have to distinguish the directly measurable quantities outside the material sample, i.e. the intensity of incident laser light, ~, and of scattered light, Is, the scattering angle, <P, and the solid angle, dQs' from the corresponding quantities inside the scattering sample, Ii' Is, ifJ, and dQs' respectively. The basic physical information is contained in I s( ifJ) which is independent of the sample's geometrical properties. Therefore, one usually converts the directly measured experimental data, Is(1)) into the corresponding quantities inside the sample. This conversion, in general, requires the knowledge of

1) the sample shape, 2) the refractice-index tensor, 3) the geometry of the light beam, and 4) the orientations of the crystal symmetry axes relative to the light beams. For a detailed description of this conversion procedure the reader is re-

ferred to LAX and NELSON (1974), and to HAYES and LOUDON (1978). In the following we generally shall use the inside-sample quantities for the description of scattering experiments.

An analogous situation holds for infrared absorption. Again, we need a conversion of the directly measured quantities, i.e. transmission, reflection, absorption, and beam geometry into quantities which do not depend on the crystals geometric properties, etc., such as the absorption constant or the complex dielectric susceptibility. For details, the reader is referred to the handbook article by E. BELL (1967).

f3) Scattering cross section and related quantities. The light scattering cross section may be expressed in terms of the (effective) interaction potential between the probing light beam and the target system. An equivalent formulation uses the dissipative part of a generalized susceptibility. Both descriptions express, in a related way, the fact that the fluctuations as induced by the perturbing light beam result in a dissipative response of the target system (fluctuation-dissipation theorem).

F or explicit expressions, let us introduce the following notations: (1) Ii) == Iq i' Wi' e): state of the incident photon with wave vector q i' energy

nwi , and polarization vector ei ,

Sect. 7

T

light beam

Theory of interaction of photons with particles

absorbed phonons anti-stokes process

I i -'

W, ,qi

target system volume V

Fig. 7.1. Light scattering due to phono

unscattered light

79

(2) In): initial (multiphonon) state of the target crystal with energy En =hwn,

(3) Is) == Iqs' w., e.): state of the scattered photon with wave vector qs' energy hw., and polarization vector es'

(4) 1m): final state of the target crystal with energy Em=hwm ,

(5) Veff : the (effective) interaction potential between the incidentcphoton and the target system. The detailed form of this potential depends on the order and the type of interaction, e.g. dipolar electron-photon interaction, onephonon Frohlich interaction, etc. The determination of Veff for the different cases is described below.

These notations correspond to the following approach (refer to Fig. 7.1): We assume that initially the complete system consists of nearly monochro

matic photons with energy Ei = hwi , wave vector qi' and transverse polarization vectors e;(q;) ; phonon states may be occupied in accord with the (low) temperature of the crystal, while electrons are assumed to remain in their ground state. Thus the initial state is

10) = li)ln). (7.1)

After a scattering process we describe the system by a final state If)

If) = Is)lm). (7.2)

The difference in energy, momentum and polarization between 10) and If) is given by the conservation laws:

(7.3)

(7.3')

and a corresponding selection rule for the polarization which involves the scattering geometry (see Sect. 13). Here Q and Q denote the net energy and


wave vectors of the multi-phonon state. Q>O and Q<O in (7.3) describe Stokes and anti-Stokes processes.

In classical electromagnetic theory the differential cross section da/dQs is the energy scattered into solid angle dQs per unit incident energy per unit area. Thus, it has the dimension of area/space angle=(cm 2 ster- 1). The geometric part of the cross section area can be visualized by inspection of Fig.7.1. It essentially means the ratio of the scattering volume ~ (i.e. that volume which is involved in the measured scattered light) and the width L of the incident light beam. On the other hand, da/dQs does not depend on the total volume V which is merely a convenient normalization factor.

It is often assumed that the incident light covers the sample volume completely, i.e. V=~. For convenience, we shall follow this custom in the present article since it does not cause any problem for sufficiently large samples.

To obtain a quantity independent of the scattering volume ~ (thus avoiding the arbitrariness in its definition) one often defines (LOUDON, 1965) a scattering efficiency S, i.e. a specific differential cross-section

1 da S=-

- ~ dQ,' (7.4)

S is a particularly useful definition in discussions of absolute cross sections (refer to CARDONA, 1982).

Another interesting point is the relation of the cross section to the intensity of emitted radiation from a harmonic dipole oscillator. This analogy suggests a w~ law for da/dQs which we find in all formulas for off-resonant scattering cross sections. We mention that this definition of da/dQs means a classical power cross-section. In present light scattering experiments very often the measuring of intensities is replaced by photon counting where photons appear as quantum particles with momentum liq and energy liw. Since every incident photon has an energy which is different from that of the scattered photons by a factor w/ws a calculation of the quantum cross-sections, based on the rate of scattered particles, exhibits a proportionality factor Wi w; instead of the classical w~ law.

In the following, usually quantum cross sections are used. da/dQs is proportional to the rate of scattered particles which is defined by

the sum over all transition rates, lili;s' contributing to a scattering from the initial state Ii) to the final state Is) into dQs ' It is given, in the first Born approximation, by Fermi's golden rule:

lili;s = 2lin I p(En)l<mqsWefflnq)12 <5 (En - Em -IiQ). n.m

(7.5)

Here, p(En} = exp( - En/kT)/Z is the temperature-dependent probability of the initial target states. The summation over the moments of final photon states is usually converted to an integration

t -+ (2:)3 S dQsdqsq;· (7.6)


The cross section is inversely proportional to the incoming photon flux, vJV where Vi is the light velocity of incident photons in the target system. For an isotropic medium with refractive index n we have the relation

V=OJ/q=c/n, (7.7)

while for anisotropic systems (7.113) has to be used. Thus we obtain finally for the spectral or partial differential cross section of isotropic crystals

(7.8)

The transition rate Vl!;s, (7.5), is related to the time auto-correlation function of veff. This can be shown by using the Heisenberg representation

with H being the crystal Hamiltonian, (7.89), and the Fourier transform

together with

Then

and

1 +00

J(OJ) =- J dt eirot• 2n -00

1 + 00

V 2 Vl!;s = h2 J dte iQt < Veff(Q, t) Veff(_ Q, 0» - 00

v 2 n n + 00 _1_' _s q2 J dteiQt<Veff(Q t) Veff(_Q 0»

(2n)3 c2 h2 8_ 00 ' ,.

(7.9)

(7.10)

(7.11)

(7.12)

(7.13)

Here, < ... ) denotes the thermal average at temperature T over all initial states with E=En'

In (7.13) the scattering cross section is given by the Fourier transform of the thermal time correlation function of the interaction potential.

')') The dynamic form factor S(Q, OJ). In the analysis of a scattering experiment it is useful to distinguish between one part which originates from the properties of the probing particles (photons, neutrons) and their interaction with the system and another part which describes the excitations of the system themselves. This second factor in the scattering cross section is usually called the dynamic form factor.

To be specific let us consider the case of a pair interaction potential between the scattering particles and the crystal

N

veff(r)= L v(r-rL )

L= 1

= J d3 r' v(r - r') p(r'),

(7.14)

(7.15)


where the crystal density is given by

(7.16)

One then obtains with (7.15) and (7.10)

verr (Q) = v( Q) p( Q). (7.17)

Inserting this result into (7.13) the cross section reads

d~d(J =qs Iv(-QWII<m'lp(Q)ln)1 2 (j(w j -Ws+Q) u; W qj m'

== qs Iv(QW S(Q, Q). (7.18) qj

Equation (7.18) defines the dynamic form factor S(Q, Q). Using the time representation of (7.11) one obtains the Fourier transform

S(Q, t) = <p(Q, t) p( - Q, 0». (7.19)

The dynamic form factor is therefore the spectral weight of the density fluctuation, i.e. the Fourier transform of the density-density time correlation function. Generally, we can calculate S(Q, Q) from the Green function, and we shall discuss the Green function method for Raman scattering in detail in Sect. 9. In this case the interaction is that of the vector potential with the current density (which involves the current-current correlation function) or that of the electric field with the dipole density (which leads to the density-density correlation function). Both interactions are equivalent via the electromagnetic gauge invariance (see the next subsection).

b) Gauge invariance in electromagnetic interaction

a) Expansion theory. In order to exhibit the electromagnetic properties of a charged particle such as an electron, which may be described by a Dirac equation, one normally uses a Foldy-Wouthuysen representation of the Hamiltonian (refer, e.g., to SCHWEBER, 1961; ZYBELL, 1967) in order to eliminate the positional degrees of freedom (particle conservation) and to obtain a systematic expansion of relativistic effects. In the presence of interaction between light and matter the Hamiltonian then becomes a series expansion in powers of the Compton wavelength of the particle, which reads to order (li/mc)2

(7.20)

Here we have neglected the energy of the rest mass, spin-dependent terms, and also the so-called Darwin term proportional to div E since it becomes important only for wavelengths shorter than 10- 10 cm. Her describes an electron in


some potential V and interacting with the radiation field determined by the potentials A and cpo

The commutation rules between p and r and between different components of (A, cp) are the usual ones for free fields (refer to BORN and HUANG, 1954).

The pure radiation part is given by the Hamiltonian (here we use an isotropic medium with 8= 1)

Hr=81n S d3 r(E2+B2). (7.21)

To make the connection between the electromagnetic fields (E, B) and the potentials (A, cp) unique we choose the radiation or transverse gauge

div A =0, cp =0. (7.22)

We then obtain 1 a -i

E= ---A=-[H ,A] e at he r

(7.23 a)

B=curl A, (7.23b)

The total Hamiltonian is the sum of Her and Hr and can now be divided into three parts:

Htot = H r + Her == H r + He + Hint· (7.24)

1 H =_p2+V

e 2m ' (7.25)

e e2

Hint = --A·p+-2 2 A 2 , me me

(7.26)

with

[p, A] =~div A =0. (7.27) I

Finally, we define the gauge-invariant velocity operator

(7.28)

We note that the matrix elements of the momentum operator p are not gauge invariant. As a consequence, the A 2 term in (7.26) can only be neglected, in contrast to what is often assumed, if we deal with processes where no momentum exchange between particles and radiation occurs, for example, if the phonon momentum may be neglected (dipole approximation, see below).

This leads us to a description of the combined system of electrons and nuclei in a crystal interacting with the radiation field. Before doing this we discuss a few more points for the simpler case of a single electron. The results can easily be generalized to the more complex case of a crystal (refer to Sect. 7e).


fJ) Dipole approximation. Let rz be the position operator for the l-th electron in the system. Its density operator is given by

p(r) = <5(r - rz)

and its particle current operator without fields by

j(r) =_1_ [pz<5(r - rz)+ <5(r- rz) pzJ. 2m

(7.29)

(7.30)

In an electromagnetic field, the true (gauge-invariant) particle current density operator becomes

e l=j--Ap

me (7.31)

which satisfies the continuity equation

:tP+diV 1=0 (7.32)

and gives the interaction Hamiltonian equivalent to (7.26)

(7.33)

Equation (7.32) shows that the gauge invariance is important to ensure charge conservation of the system.

We are now going to show that (7.20) can be transformed into a form which is explicitly gauge invariant for the most important case where the wavelength of the light is large compared with the local dimensions of the crystal and relativistic effects can be neglected. This transformation is well known from the classical treatment by GOEPPERT-MAYER (1931). It is most easily described by starting from the classical Lagrangian for a particle with charge e, which is equivalent to Her' Eq. (7.20),

1 '2 e. Ler=Imr +-rA-V.

e (7.34)

The idea is to replace the gauge-dependent vector potential A by the gaugeinvariant electric field E which is essentially the time derivative of A. One adds

therefore the total time derivative -~ dd (r· A) to L which does not change the e t equations of motion and obtains

, 1 '2 e d L =-mr --r-A- V er 2 e dt ' (7.35)

Sect. 7 Theory of interaction of photons with particles

which corresponds to the classical Hamiltonian

, 1 2 e d Her =-2 p +-r-d A + V.

met

85

(7.36)

In the long-wavelength limit where A has a weak dependence on r we may replace

d dt A(r, t)

a by at A(r, t) = - eE(r, t) (7.37)

(dipole approximation) so that the interaction Hamiltonian (7.26) becomes

(7.38)

The advantage of (7.38) is that the dipole moment of the particle, e r, is gauge invariant as is the electric field E.

We shall now see that the same result is obtained by a gauge transfor-

mation. The procedure analogous to the addition of : (r A) to L is the change of the gauge for A and cp: t

with the gauge function

A'=A+VX

, 1 a cp =cp---x

eat

X= -rA.

(7.39)

(7.40)

(7.41) We obtain with V· A =0, cp =0

A'=A -V(rA)= -{rV)A -r x (VX A)= -(r grad) A -r x B (7.42)

, 1 a cp =+-;;ratA=-r.E, (7.43)

and for the interaction Hamiltonian (7.33)

H;nt=-Jd3repr·E (7.44)

+ J d3 r [~j. «r grad) A + r x B)] (7.45)

e2

+Jd3r-2 2[p«rgrad)A+rxB)fJ. (7.46) me

H;nt contains, successively, the electric-dipole interaction (7.44), the electric quadrupole and magnetic dipole interactions (7.45), and higher order corrections (7.46). The dipole approximation neglects all but the first term in Hint which reads for a single particle, cf. (7.44),

(7.47)


with the dipole operator (7.48)

The time dependence of E in (7.47) is assumed to be 'slow', i.e. the photon energy hco has to be small compared with the electron rest mass mc2 or

(7.49)

which is again the above-mentioned criterion that the wavelength A of the exciting light, in the non-relativistic limit, has to be large compared to the Compton wavelength Ac' The dipole approximation is automatically obtained if the external field is nearly homogeneous in the region of interaction. It means that in the plane-wave representations for A and E all wave vectors q can be assumed to be equal to zero. Since the electric quadrupole, the magnetic dipole and the higher-order terms contain spatial derivatives of A, they are proportional at least to the first power of q and therefore disappear. The only term to be retained is then again the dipole term (7.44), where now the electric field in (7.47) becomes a spatial constant. This is the simplest form of the dipole approximation.

The gauge defined by (7.41), (7.42) and (7.43) may be called the dipole gauge. Here VA' oc(q· A') -+ 0 means that the transverse gauge is still retained. This does not hold for the higher order terms (7.45) and (7.46).

y) M ultipole expansion of interaction. The dipole gauge provides a complicated division of the A 2 term into different contributions to dipole, quadrupole, etc., terms in H;nt. It is interesting to transform the higher-order terms in (7.45) and (7.46) to a form where all multipole terms are given in an explicitly gauge-invariant representation, i.e. only using the gauge-invariant current J(r), the electromagnetic fields E and B, etc. As can be seen by a discussion of (7.42) and (7.43) this can be done by a generalization of the dipole gauge to a "multipole" gauge which contains a sort of a Taylor series expansion in spatial derivatives of A. The problem has been discussed for the case of a single atom or molecule by several authors (refer to LOUDON, 1973). Generally, the electric quadrupole and the magnetic dipole interactions are smaller than the electric dipole interaction by a factor of the order of the fine structure constant, (X~137-\ and become important only if the transition under consideration is dipole forbidden. A general discussion for a system of charged particles has been given by FIUTAK (1963) where the difference between a classical and a quantum-mechanical description of the external radiation has been pointed out. In molecular spectroscopy the different equivalent representations of the interaction operator are often described in terms of 'dipole-velocity' expressions, if the momentum p is used, or' dipole-length' expressions, if the coordinate vector r is chosen. For a discussion of these and other definitions and their importance in molecular physics refer to BALLHAUSEN and HANSEN (1972).

c) Dielectric constant of electrons: Dispersion relations and sum rules. The dielectric tensor caP(q, co) for the response of the electrons in a crystal to an


external electromagnetic radiation field is given (if spatial dispersion can be neglected, i.e. q~O) in terms of the transition matrix elements of the dipole operator (7.48)

m=Imz=eIy,==eR, (7.50) , , by

Sap(w) = bap + 4n Xap(w) (7.51)

(refer to (6.81) for w = 0) with the transverse electronic susceptibility

e2 1 XaP=1'_VI<nIRalm) <mIRpln) _( +.)

fI m wmn W IS (7.52)

The In) are many-electron eigenstates. The electrons are assumed to be in their ground state. The transition frequencies wmn correspond to the excitation energies from the ground state,

hwmn=Em-En· (7.53)

Here we need no specification of the dipole matrix elements

(7.54)

We only assume that boundary conditions of the system are chosen to ensure that the Rnm are well-defined quantities. 1

If we are dealing with an isotropic (e.g. cubic) system, I: reduces to a scalar so that the real and imaginary parts of S may be conveniently expressed by the refractive index n and the absorption constant k:

s(w) = Re s(w) + 1m s(w)

== s'(w) + i s//(w) (7.55)

= n2(w) - k2(W) + i· 2n(w) k(w).

We note a few general properties of s(w):

C() Dipole sum rule. The imaginary part of the dielectric constant, s//(w), fulfills an important sum rule:

(7.56)

where N is the total number of electrons and wp the so-called plasma frequency (refer, e.g., to BA YM, 1969).

In (7.56) we have used

(7.57)

which is called the dipole sum rule originally found by THOMAS (1925), REICHE (1925) and KUHN (1925) for atomic transitions. Every transition with

1 This is always the case for a finite crystal. The problems connected with periodic boundary conditions or infinite crystals are discussed in Sect. 9.


transition energy hWnO contributes to the sum with a certain oscillator strength (for scalar 8)

f. _2mwmn 2

mn= 3h Rmn· (7.58)

Equation (7.57) then reads (7.59)

n

and is called the f-sum rule. Instead of using the dipole operator m we could have started from the non

gauge-invariant interaction Hamiltonian (7.26), which is the usual procedure. We denote the total momentum operator of the electrons by

P= LPI. I

With the help of the commutation relation

hpl= -im[r"HeJ

we obtain the matrix elements

p..m=<nl P 1m) = -imwnm<nl Rim)

which gives the electronic susceptibility

( )=~ ~ P..mPmn 1W L. 2 .• hV m Wmn(Wmn -(W+18))

(7.60)

(7.61)

(7.62)

(7.63)

(7.64)

The matrix elements P..m are not gauge invariant but change by an extra term under the transformation (7.41)

e - <nl grad X 1m). c

(7.65)

However, in the dipole gauge (7.41) this term would exhibit only electric quadrupole and magnetic-dipole contributions which disappear in the dipole approximation. Therefore, for the case of dipole absorption the two treatments are equivalent.

/3) Dispersion relations and sum rules. The dipole sum rule is only the most important example of sum rules for the optical constants of material media which may be derived from dispersion and super convergence relations. This derivation of sum rules has been investigated recently by ALTARELLI et al. (1972, 1974), and we follow their discussion in this subsection. For a recent review see SMITH (1982).

The interest in dispersion relations and sum rules is motivated by the fact that optical constants are measured over a necessarily limited range of frequen-


cies so that extrapolations, Kramers-Kronig inversions and sum rules are required for the complete analysis of optical data.

The main point in the derivation of sum rules is the observation that electronic systems such as in atoms or in solids respond to an external field in the same way as a system of an equal number of free electrons if the frequency is much higher than any of the resonances. This leads to a rapid fall-off of the various optical constants as the frequency approaches infinity. This rapid convergence of the frequency integrals corresponds to the causal behavior of the system, in particular, to the impossibility of an instanteneous response, i.e. to its inertia. One may express this behavior by a 'superconvergence theorem' which seems to be the most powerful analytical technique for the derivation of sum rules. We do not go into the details of the discussion by ALTARELLI et al. but merely summarize some important results including those sum rules which are well known.

In an isotropic medium, the Kramers-Kronig relations (refer to (34.83)) for the complex refractive index Ii = n + i k, Eq. (7.55), read

2 00 OJ' k( OJ') , n(OJ)-l=-PI '2 2dOJ

n 0 OJ -OJ (7.66)

and k(OJ) = -2OJ P OOI n(OJ')-1 dOJ'.

n 0 OJ'2 -OJ2 (7.67)

The requirement that the response of the medium IS like that of a free electron gas in the high-frequency limit, means that

2nNe2

Fi( OJ) - 1 ~ - -m-V-OJ-"'2 OJ --> Cf). (7.68)

Furthermore we obtain the following sum rules which have been extended to the anisotropic case:

1) the original f-sum rule (7.59), 00

I OJ 1m 8ap (OJ) dOJ =~nOJ~ rSaP ' (7.69) o

2) the corresponding rule for the inverse dielectric constant 00

I OJ 1m 8;/ (OJ) dOJ = -~nOJ~ rSaP ' (7.70) o

3) the f-sum rule for the isotropic absorption coefficient 00

I OJk(OJ) dOJ =tnOJ~. (7.71) o

In addition, sum rules exist which are specifically connected with the inertia of the system

00

4) I [n(OJ)-l] dOJ=O (7.72) o


00

5) S [ReB"p(w)-(\,pJ dw= -21Ca"p(0), (7.73) o

where a"p(O) is the ap-component of the conduction tensor (i.e. equal to zero in insulators), and

00

6) S [ReB,;/(w)-b"pJ dw=O. (7.74) o

Very recently a new sum rule has been given by Wu et al. (1976) which, in contrast to the short-time response connected with dipolar sum rules, is connected with the long-time response. This quadrupolar sum rule reads

(7.75)

The results for the electronic absorption given here can easily be transferred (with a slight change of notation) to the case of ionic dipole and quadrupole absorption. We shall see this in Sect. 8 where the infrared absorption is treated for the general case of an ensemble of interacting oscillators in a thermal bath.

d) Light scattering by electrons. In the foregoing section we described the interaction of light with electrons to lowest (second) order in the coupling constant e. The result is the absorption or dispersion of light inside the crystal.

In the next order, proportional to e4, we allow for a change of wand q of the photons due to scattering by the electrons. Again we neglect effects of non-zero wave vector. The differential cross section for light scattering can be determined from the so-called transition polarizability amplitude (refer to BORN and HUANG, 1954, p.203)

p( . ) =~ ,,[ Rnm, Rm'n Rnm, Rm'n ] ~~ kL . + .

ft m' wnm,+Wi+lB wm'n-wi+1B (7.76)

with (7.54) and the notation of Sect. 7 p. The differential cross section for an element of solid angle dQs is then given

by da(w;. ws) e4 31 pnm( )1 2

d 4WiWs es Wi,Ws es' Qs c

(7.77)

For m=n we obtain the low-frequency cross section for elastic Rayleigh scattering (Wi =Ws)

(7.78)

For comparison with (7.77)ff. we note that the differential cross section for a system of N free electrons, as derived from <nl A2 In), takes the highfrequency form

da(wi' ws) 2 2 Ws 1 12 N r - e e· dQ 0 w. s 1 S 1

(7.79)


which is N 2 times the so-called Thomson cross section of a single electron where ro = e21m c2 is the classical electronic radius. At long wavelengths the excitation of internal electronic degrees of freedom of an atom, molecule or small crystal is negligible. The system then shows a behavior as is given in (7.79) where the N electrons coherently scatter the light as if they were one large single particle with charge N e. Here, phonon-like excitations are not considered (see Sect. 9).

We can now see that for the description of light scattering by localized systems the use of a Hamiltonian which is gauge-invariant is advantageous and leads immediately to the correct cross sections. Sometimes, as is often done in resonant Raman scattering, one prefers to use the matrix elements of the momentum operator p which have a simple meaning in the usual type of band theory. Then, one has to be cautious with the representation of the cross section if the correct frequency dependence cx:w4 should be retained (refer to Sect. 9).

e) Interaction of photons with electrons and ions. In the foregoing sections we have described the electrons in a crystal by an effective local potential V. We now proceed to an explicit description of all particles in the crystal and begin with the non-relativistic Hamiltonian of the total system of radiation and matter

(7.80)

where the Hamiltonian of the radiation H r in the radiation gauge is given by (7.21) and the gauge-invariant Hamiltonian of the crystal including its interaction with radiation reads

H m=,T.(A)+7;(A) (7.81)

+Ue+Uj+Uej · (7~82)

Here,

1 ( e r T(A)=-L p --A • 2m I I C

(7.83)

and

1 ( eZ r 7;(A)=L 2M Pn----;-A (7.84) n n

are the gauge invariant kinetic energies of the electrons and nuclei, respectively. The charges eZn are the bare charges of nuclei, in principle, but we shall later prefer to replace them by effective static core charges and to modify the interaction terms correspondingly. The terms denoted by U are

the electronic interaction,

the ionic interaction, and

1 e2

U.=-L'--2 1",I.rI -rl'l

and that between electrons and ions

(7.85)

(7.86)

(7.87)


e e The crystal excitations with gauge invariant momenta pz -- A and Pn -- ZnA

c c are the so-called (excitonic and phonon-like) polaritons. We shall discuss these quasi-particles in Subsect. 7f in the harmonic approximation where their introduction is particularly useful. For the general case we prefer to describe the excitations in a crystal without the electromagnetic field. We therefore divide Hrn into a crystal part and an interaction term (cf. (7.26) and (7.27)

(7.88) with

(7.89)

Hint=+ ~ L [-A Pz+ 2e A2]

mc z c (7.90)

e" Zn [ eZn 2] +-L-- -APn+- A . C n AIn 2c

(7.91)

Note that the individual terms in the Hamiltonian Hint are not gauge invariant and, therefore, do not give matrix elements which are observables of the crystal. Since for most cases we can use the simple dipole approximation, we transform Hint with the help of the (generalized) dipole gauge, Eqs. (7.41)-(7.43), into a more convenient form. With the introduction of the charge density

e per) = eeL b(r - rz) + L Znb(r - rn)] z n

and the charge current density

. e eJ(r) =-2 L [Pz b(r - rz) + b(r - rz) pz]

mz

+ ~ L ~n [Pn b(r - rn) + b(r - rn) Pn] n n

(7.92)

(7.93)

we formally obtain the result of (7.44)-(7.46) for Hint. The dipole approximation then gives

(7.94) with the dipole operator

mer) =e r per) (7.95)

and the external field E. We remark that higher-order (electric quadrupole etc.) terms can be obtained

from (7.44) and (7.46) if necessary. The foregoing discussion shows that the full quantum-mechanical treatment of the coupled system of radiation and matter in the dipole approximation leads to the same result as the semi-classical treatment (refer to FIUTAK, 1963) and Sect. 7bf3.

f) Polaritons in the harmonic approximation. If the crystal potential is restricted to the harmonic approximation then the Hamiltonian H of the complete system 'radiation and matter' is bilinear in the dynamic variables and


can therefore be diagonalized exactly. The new degrees of freedom are then mixed photon-polarization modes which have first been introduced by HUANG (1950) and later called polaritons by HOPFIELD (1958) for the case of excitons. In the following we shall adopt this name also for phonon-polaritons. The results for this case can easily be transferred to those of exciton-polaritons.

The dispersion relation ws(q) of these quasi-particles of the combined photon-phonon system in the harmonic and dipole approximation have been discussed for cubic crystals by BORN and HUANG (1954) and for uniaxial crystals by LOUDON (1965); BURSTEIN et al. (1970); and MERTEN (1967, 1968). For a recent review see CLAUS et al. (1975), CLAUS (1980). They can be measured directly by Raman scattering. The theory of this scattering was first given by BURSTEIN et al. (1968). For further references refer to SCOTT (1971) and HAYES and LOUDON (1978).

First we recall the well-known case of a diatomic cubic ionic crystal (BORN and HUANG, 1954). The equations for linearly polarized harmonic polaritons may be obtained by an extension of the description of long-wavelength optic phonons to the retarded case.

The Maxwell equations for photon-phonon waves "-'exp(i[q y-w tJ) are (1 = longitudinal, t = transverse)

div(E +4nP) = i q(E1 +4npI ) =0, (7.96)

div B1=iqB1=0 (7.97)

1. w curlE+-B=iq xEt-i-Bt=O

c C (7.98)

and 1.. w

curlB--(E +4nP)=i q x Bt+i-(Et+4npt)=0. (7.99) c c

Equations (7.98) and (7.99) give

(7.100)

or

(7.101)

Equation (7.96) shows that the situation is unchanged for longitudinal waves since no retardation effects exist in this case. Therefore, one obtains unchanged values of WLO in the whole regime for which cq;5WLO. Another way to express this result is to say that there is no coupling at all between an external (necessarily transverse) electromagnetic field and the purely longitudinally polarized optic phonons in a crystal. Therefore, the polariton concept is restricted here to transversely polarized excitations. Equation (7.101) gives the dispersion relations for the polaritons in terms of the frequency-dependent dielectric constant:

(7.102)

94 Interaction of photons with matter

500

400

1 u 300~--~~---7/~-------------WLO ~ ---------7t-------------- WTO

200

100

a

h h

I /

I.

I

/ I

/

I

I /

/ /

q. (10 4 cm-1 )

Sect. 7

Fig. 7.2. Polariton dispersion curves in GaAs, schematical (after HAYES and LONDON, 1978)

We compare this with the well-known expression for the dielectric constant of a diatomic cubic crystal with an effective transverse charge ZTe, reduced mass fl, high-frequency electronic part GOO' and resonance frequency WTO given in (6.66). This equation can also be obtained from (7.51) by appropriate replacements of the electronic parameters by those of an ionic oscillator.

Inserting (6.66) and the Lyddane-Sachs-Teller relation into (7.102) we obtain for the transverse vibrations:

[ q2 C2 ] wlo q2 c2

W 4 _W 2 wlo+-- +---"''----Goo Go

O. (7.103)

This quadratic equation for the squared transverse frequencies wi leads to two different regimes of solutions which are shown in Fig. 7.2. The low-frequency regime (0 < W T ~ WTO) is characterized by c q ~ WLO and

c2 q2 wi=--

Go (7.104)

In the high-frequency regime, w::::: wLO' the result for small wave vectors (q~WLO/C) is:

giving wi=wlo for q-+O,

while for large wave vectors 2 2 2 c2 2

WT=WLO-WTO+-q Goo

for q '?> wLO/c.

(7.1 05)

(7.1 06)

(7.107)

(7.108)

Sect. 8 Infrared absorption and dielectric response 95

Equation (7.106) shows that W T and WLO are degenerate for q--+O as is required by 0h symmetry. Equations (7.104) and (7.108) express the relation between the refractive indices (no and n",,) and the transverse optic frequency in different frequency regimes.

In the regime W~WTO' we obtain from (7.103):

2 2 ( eO - eoo 2) WT=WTO l-~WTO

--+wio for q~wTOle.

(7.109)

(7.110)

Equation (7.109) describes the regime between the polariton regime (q~wTOle) and the non-retarded phonon regime discussed in Sect. 4e.

The polariton dispersion relation (7.109) can be measured in Raman scattering experiments at small scattering angles. This was first demonstrated by HENRY and ROPFIELD (1965).

These results can be generalized for crystals with lower symmetries and more than two particles in the elementary cell. In the limit w--+O we obtain from (6.67) the generalized Lyddane-Sachs-Teller relation (COCHRAN, 1959; COCHRAN and COWLEY, 1962; KUROSAWA, 1961) for cubic crystals with 3r optic branches

(7.111)

The polariton dispersion relations are derived by combining (6.67) with Fresnel's equation for the wave normals:

222 Sl S2 S3_0

-1--1 +-1--1 +-1--1 - (7.112)

n2 -;-; n2 - e2 n2 - e3

where sa. is a direction cosine and n = q elw the refractive index. The polaritons can be divided into ordinary polaritons with purely trans

verse polarization and orientation-independent dispersion branches, and generalized polaritons with mixed transverse and longitudinal polarization and orientation-dependent branches.

The problem of the linewidth of polaritons will be discussed in connection with the theory of Raman scattering in Sect. 13.

8. Infrared absorption and dielectric response. After the outline of the general theory in Sect. 7, our discussion begins with the infrared absorption and describes this as the complex transverse dielectric response of the crystal to a (classical) external electric field. The situation is very reminiscent of a system of one or more oscillators with frequency-dependent damping moving in a forced vibration with a single applied frequency. In the simpler crystals usually only one or very few dispersion oscillators exist. This leads to a rather simple structure of the spectra in terms of a few reflection or absorption bands. The interesting features appear in the secondary structure of these bands whieh give detailed information about the anharmonic coupling of phonons and related properties.


There is, therefore, no advantage in transforming, at the outset, the total Hamiltonian into a system of many coupled phonon-photon modes (phononpolaritons) which all individually contribute to the absorption spectra because then the useful resonance-oscillator concept is lost.

The theory of infrared absorption is presented here as a straight-forward first-order quantum-statistical perturbation theory of a phonon ensemble at thermal equilibrium (Kubo-formula). This microscopic approach to the linear response of a system to an external field has to be complemented by (and to be consistent with) the macroscopic thermodynamical theory of susceptibilities which defines these quantities in terms of certain derivatives of thermodynamic potentials with repsect to 'external' fields. The fact that both procedures usually lead to the same result is by no means trivial but intimately connected with the problem of a microscopic statistical foundation of the thermodynamics of solids. For a detailed discussion the reader is refered to standard text books (KADAN OFF and BAYM, 1962; MONSTER, 1969, 1973). A few important points are summarized at the beginning of this chapter.

a) Dielectric susceptibility. The dielectric response of a crystal to an external electric field E is the induced polarization P. It is represented as a function of the applied field by the (linear) dielectric response function or susceptibility x. In the case of a static homogeneous field, X is defined by the relation

1 p=V {M[EJ-M[E=OJ}=X· E. (8.1)

Here, M denotes the dipole moment of the crystal with volume V. In the more general case of an electromagnetic wave, the response depends on wave vector and frequency. This dynamic response includes the effect of absorption. In this section, the radiation field is treated as a classical field and the magnetic component is disregarded. In that case only a space-dependent oscillating electric field, which is switched on adiabatically and interacts locally with a dipole moment density m(r), needs to be considered. The corresponding interaction Hamiltonian (7.94) was defined in the dipole approximation. The dipole moment density is given as a sum over all nuclei (n) and electrons (1) of the crystal, cf. (7.92)-(7.95):

mer J = e rO: Zn 6(r- rn) - L 6(r - r1)]. (8.2) n 1

In a shell model this quantity is expressed in terms of the core and shell charges (refer to Sect. 14).

Defining the space Fourier transforms as

1 '\' . m(r) =- L. m(q) e,qr V q

m(q)= S d 3 rm(r)e- iqr v

m(q=O)=M

(8.3)

(8.4)

(8.5)


the interaction Hamiltonian of (7.94) is rewritten in the form

(8.6)

In the static homogeneous case this expression reduces to

H~iP= -E· M. (8.7)

The dielectric response is determined by calculating the expectation value of the dipole moment. For this the quantum mechanical theory of linear response at finite temperature as developed by KUBO (1975) is used. In the absence of an externally applied field the static dipole moment of the crystal is given by the thermodynamic expectation value

M[E=O] =<M) =Tr{pM}, (8.8)

where the density matrix p, corresponding to thermal equilibrium, is given by

(8.9)

with Z=Tre- PH and {3=l/kT. H is the Hamiltonian of the crystal in the absence of a radiation field, (7.74).

As a result of the external radiation, the density matrix and the expectation values become time-dependent. In the Heisenberg picture, the expectation value of the dipole moment is represented as

<m(q, t)=Tr{p(t) m(q)}, (8.10)

where the density matrix p(t) is the solution of the equation of motion (TER HAAR, 1961)

iii :t p(t)= [H + H dip ' p(t)]. (8.11)

The linear response is defined by the deviations of p(t) and of the dipole moment from their equilibrium values to first order in the perturbing electric field

and p(t)= p + Llp(t),

<LI m(q, t) = <m(q, t) - <m(q),} = Tr{LI p(t) m(q)}.

(8.12)

(8.13)

Using (8.12) in the equation of motion of the density matrix and with the initial condition Llp( - 00)=0, the solution (KUBO, 1957) is found to be

1 t -~H(t-t) ~H(t-t) Llp(t)=ili S dee h [Hdip,nP]eli .

-00

(8.14)

From this, the induced dipole moment follows as

1 00

<Llm(q, t) = iii S de e(t-e) <[m(q, t-e), Hdip,tJ) (8.15) -00

98 Interaction of photons with matter

where m(q, t) is the dipole-moment operator in the interaction picture

~Ht -~Ht m(q, t) = eh m(q) e h

Furthermore, B(t) is the unit step function

B(t)={O, 1,

get) = (j(t).

t<o t>O,

Inserting the interaction Hamiltonian (7.94) into (S.15) gives:

<,dm(q, t) = lim -hi L 7 dT B(T)<[m(q, T), mfJ(q', 0)]) e-O V fJ.q' -00

x ei(w+ie), EfJ( _ q') e- iwt.

Sect. 8

(8.16)

(8.17)

(8.1S)

The susceptibility tensor can now be defined in a more general way than in (S.l), that is

<,dma(q, t)= L XafJ(q, q', w)EfJ( -q') e- iwt. (S.19) fJ. q'

This leads to 1 00

XafJ(q, q', w) = lim hV S dt B(t)<[ma(q, t), mfJ(q', 0)]) ei(w+ie)t. (S.20) e~O ~ 00

Here, the susceptibility is expressed as the Fourier transform of the temperaturedependent retarded Green function (Sect. 34c)

(S.21)

Since we are dealing here with this type of Green function only, the superscript r used in Chap. G is omitted. With the Fourier transform defined by

we put (S.20) into the form

00

G =lim S dtG eiwte-eltl wI'

10-0 - 00

G 1 OOs d G -iwt 1=-2 w we , n -00

XafJ(q, q', w) = h ~ Gw(ma(q) I mfJ(q'))·

(S.22)

(S.23)

(S.24)

So far, the interaction ofthe radiation with the crystal has been treated without discussion of the internal structure of the crystal. Expressed in the form (S.24), the susceptibility can be calculated when the dipole moment m(q) and the Hamiltonian H are specified. Therefore, the treatment contains the absorption via the ionic (Reststrahlen) oscillators (cf. diagram a in Table 2) as well as that by the electronic dipole moments which, in cooperation with the non-linear electron-ion interaction lead to the so-called non-linear dipole moments in homopolar crystals like Ge (cf. diagram b in Table 2).


b) Absorption of radiation (fluctuation-dissipation theorem). The dielectric susceptibility is characterized by general properties which follow from basic physical assumptions. Extensive treatments are given by STERN (1963) and AGRANOVICH and GINZBURG (1966). For the purposes of these sections it is helpful to write the susceptibility explicitly in terms of matrix elements. The representation is naturally done in the eigenstates, n, of the isolated crystal with Hamiltonian H, (7.89). According to the definitions of the foregoing section, and with Hnn=hwn, and Pn=Z-l e- Phwn, (8.24) can be written in the form of the spectral representation of the retarded Green function (34.60)

( ') 1" Pn - Pm ( ) ( ') Xapq,q,w=-hVL... + +' maqnmmpq mn' nm W Wn-Wm IB

(8.25)

Since the dipole moment density m(r) is a hermitian operator, m(q)nm =m( -q)~n' the reality condition

Xap(-q, _q', -w)=Xap(q,q',w)* (8.26)

is verified with (8.25). For most purposes, it is sufficient to consider, in the general tensor of (8.25),

only those elements with q' = - q, i.e.

(8.27)

For example, in the case of a perfect lattice, it follows from lattice periodicity that the product of matrix elements ma(q)nm mp(q')mn is only nonvanishing if q + q' is equal either to zero or to a reciprocal lattice vector. In the infrared and even in the visible region the wavelength of the radiation is very large compared with the lattice constant, so that for a perfect crystal attention can be restricted to q' = - q ~O. If the product of matrix elements ma(q)nm mp( - q)mn is symmetric in the states nand m, one can write

(8.28)

=Xpa(-qW).

The rate of energy absorbed by the crystal is derived from the increase of the crystal energy U during a short time interval

dU = S d 3 r E(r, t)· d<m(r, t»

or, equivalently, with the aid of (8.6) and (8.13),

dU 1 d (it = V L E( - q, t)· dt <LI m(q, t».

q

(8.29)

(8.30)

In the case of a monochromatic linearly-polarized plane wave with amplitude E

E(r, t)=E cos(qo' r-w t)

with the Fourier components

(8.31)


v . . E(q t)=-E(b e-Iwt+b elwt) , 2 q. qo q. - qo ' (8.32)

according to (8.19) the change in the dipole moment becomes

V~ . . <Am,,(q, t)='2 L, {X"p(q, -qQ, w) e-Iwt+x"p(q, qQ, -w) elwt} EfJ' (8.33)

fJ

If this is substituted into (8.30), after taking the time average, the familiar result for the absorbed energy is obtained:

(8.34)

Here we have assumed that the crystal is in dynamic equilibrium with the radiation (stationary phase and amplitude). This is only the case if the energy transferred per cycle is not stored but is released as heat from the crystal into a thermal bath. In the theory this fact is built in, phenomenologically, by the quantity e in (8.28). It is in this way that the aspect of irreversibility is introduced when absorption is calculated in terms of stationary eigenstates of the isolated crystal.

The imaginary part of the dielectric susceptibility describing absorption is derived from (8.28) using Dirac's identity

lim _1_. =P!=t=i7Cb(x). (8.35) .~o+ x±le x

F or the diagonal elements (oc = f3), which in the case of cubic crystals are the only important ones, one obtains

1m X",,(qw)= h7CV (l_e- PIiW) L Pn Im,,(q)nmI 2 b(w+ wn -wm)' (8.36) nm

The right-hand side of this equation can be expressed in terms of the Fourier transform of the dipole moment auto-correlation function (refer to (34.69))

00

J dt eiwt <m,,(q, t) m,,( - q, 0) =J w(m,,(q) I m,,( - q)) (8.37) -00

=27C L Pnl m,,(q)nmI2 b(w+wn-wm)· (8.38) nm

The function J represents the spectral density of the fluctuations of the dipole moment. For finite frequencies the spectral density function follows from (8.36) as

Jw(m,,(q)lm,,(-q))=2hV[1+n(w)] 1m X",,(qw), (w =l= 0), (8.39)

with (8.40)

Since the imaginary part of X",,(qw) is an odd function of frequency and because of 1 + n( - w) = - n(w), (8.39) may be written as

J w(m,,(q) I m,,( - q)) +J _w(m,,(q) I m,,( - q))

=2hV coth (f3~W) Imx",,(qw), (w=l=O), (8.41)


where (f3 hW ) coth -2- = 2n(w) + 1. (8.42)

The relation (8.41) is the fluctuation-dissipation theorem (CALLEN and WELTON, 1951). Generally, it relates the fluctuation spectrum of any force to a quantity characteristic of energy loss.

Static correlations, which lead to a delta function at zero frequency in (8.38), are excluded from (8.41). They have attracted the attention of several authors (STEVENS, 1965; CALLEN, 1967; KWOK, 1969). The special problem of the static limit of the Kubo susceptibility is discussed at the end of this chapter.

c) Frequency-dependence and thermodynamic definitions of the susceptibility, sum rules. Various properties of the susceptibility follow immediately from (8.28) which is a general expression for perfect lattices. In this equation the susceptibility is represented by a sum of weakly-damped classical dispersion terms. The resonance factors

1 lim 2 2'

8--+0+ Wnm-W -lSW 1 . ~(2 2 ) P 2 2 +111: sgnWu W -Wnm

Wnm - W (8.43)

are the Fourier transforms of e (t) sin Wnm t . (8.44)

Wnm

This function exhibits the reaction of an oscillator with eigenfrequency Wnm to an instantaneous excitation at time t = 0. For t < 0, the oscillator is at rest; for t>O, there is harmonic motion (Fig. 8.1). The step function e in (8.44) represents the fundamental property of causality, i.e. the fact that action always precedes reaction.

From (8.28) it is seen that for diagonal elements ((I. = 13)

x(q - w) = X(q w)*, (8.45)

i.e. that the real and imaginary part of X = X' + i X" are even and odd functions of w, respectively,

x'(q - w) = X'(q w),

X" (q - w) = - X"(q w).

If these relations are combined with the contour integral

x(qW)=~ P~du X(qu) (u~O) 11:1 u-w

the Kramers-Kronig relations are obtained in the form

2 00 u X'(qw)=- P J du 2 2 X"(qu),

11: 0 u-w

2 00 W x"(qW) = -- J du 2 2 X'(qu).

11: 0 u-w

(8.46)

(8.47)

(8.48)

(8.49)

(8.50)


These relations are widely used for the determination of optical constants from experimental data (E. BELL, 1967; WILHELMI, 1968; GREENA WA Y, 1968) .

. Further spectral properties of the susceptibility can be derived from the Fourier transform of the commutator in the retarded Green function (8.21). In the diagonal case, a = /3, q' = - q, one obtains

00

J dt eiwt <m~(q, t) m~( - q, 0) -m~( - q, 0) m~(q, t) = 2h V 1m X~~(q w). (8.51) -00

This equation can be used as a starting point for discussing sum rules and moment expansions; the reader is referred to KADANOFF and MARTIN (1963) and to Sect. 7 c where all important sum rules for the electronic case have been given. They may be easily transferred to the case of an ionic dielectric response, if some simple rules are observed:

1) The high-frequency limit in infrared absorption (' cD') corresponds to frequencies where the ions are practically at rest but the electrons are still moving slowly, i.e. they are still in their low-frequency 'static' regime. Therefore, the electronic limit, C -d, has to be replaced by the ionic limit, c -+ coo' i.e. the optic dielectric constant.

2) The electronic masses have to be replaced by the corresponding ionic masses. For example, in the case of a diatonic cubic lattice, the ionic reduced mass J1 has to be used instead of the electronic mass m.

3) The electronic charge - e has to be replaced by the effective ionic charge, electronic dipole matrix elements by ionic ones, etc.

'00' 2n2 J wc"(w) dw=- (ZTe)2 o J1 Vo

(8.52)

n 2 =2: (co-c",) wTo . (8.53)

This infrared dipole sum rule is completely analogous to (7.56) if one considers that ZTe is the transverse effective ionic charge and n the number of positive charges per unit cell. The oscillator strength is essentially given by the difference between the total static dielectric response Co and the electronic static response c"'. Using the Lyddane-Sachs-Teller relation (4.7), we can write (8.53) in the following form:

, ooJ ' c" ( w ) n 2 2 n 2 dW'W--=-2 (WLO -WTO )=-2 wpL

o C'" (8.54)

n C::::2: (WLO + wTO) LJR, (8.55)

which shows that the frequency width of the reflection band, LJR, approximately given by WLO - W TO , describes the integrated absorption reduced by coo' The analogy with the electronic case is obvious if we compare (7.56) and (8.54) with the electronic and ionic plasma frequencies, respectively (refer to (4.9)).

Other sum rules for infrared absorption may be obtained in a similar way from Sect. 7 d. Of interest are asymptomic formulae like those for the refractive


index ii, (7.68), which reads in the infrared regime

(8.56)

with similar formulae for the absorption index k(w) and the reflectivity R(w). The superconvergence theorem leads to formulae like the following

, 00'

J [n(w)-n oo ] dw=O, (8.57) o

and corresponding expressions for e and R. Finally we mention the quadrupolar sum rule, (7.75), which reads in the

ionic case:

(8.58)

In the closure approximation, the polarizability of a system due to virtual dipole excitations <01 R Ii) is given by

_ ~ <01 R Ii) <il RIO) Xion - 2 L. E - E

i i 0

(8.59)

2 2 =-·<OIR 10).

L1Eeff (8.60)

Here the effective gap L1Eeff corresponds to an average phonon energy. Therefore, Eq. (8.58) connects the electronic and ionic static polarizabilities with the first negative moment of the infrared absorption, i.e.

J W-11l((w) dwoc J e"(W) dw. (8.61)

d) Static susceptibility. A point of particular interest is the static susceptibility. In the limit of vanishing frequency, the results of the dynamical theory should be in agreement with the thermodynamic theory of quasi-static processes. In thermodynamic considerations, q is usually set zero. If the direction of the electric field E is assumed along a cartesian axis, the definitions of the isothermal and adiabatic susceptibilities are simply written

where the field-dependent expectation value of the dipole moment

M[E] = _ aG[E] aE

is derived from the Gibbs free energy

1 G[E] = -p In Tr {exp [ - f3(H -ME)]}.

(8.62)

(8.63)

(8.64)


In order to evaluate this expression, ordinary perturbation theory is employed. If there are no matrix elements of the dipole moment between degenerate states, the perturbed energy eigenvalues are

hCOn[E]=hCOn-MnnE+-h1 I' IMnml2 E2+.... (8.65) m con-com

When the terms up to second order in the electric field are inserted into (8.64) the free energy is found to be

V 2 G[E]=F-(M)E-T XT E + .... (8.66)

Here, F == G [E = 0] = - f3 -1 In Z, is the free energy of the unperturbed crystal, and the isothermal susceptibility, XT is given by

(8.67)

During the quasi-static isothermal switching on of the electric field, the change of the dipole moment is, to first order in the field, given by

L1M[E]=M[E]-(M)= VXTE;

simultaneously, heat is transferred to the crystal according to

L1Q- L1S - T·

The change of the entropy again is calculated as

oG[E] of o(M) L1S= -----ar+ aT=ar- E+ ....

(8.68)

(8.69)

(8.70)

The adiabatic susceptibility is derived by requiring L1 S = 0, while the temperature is allowed to vary, then Eq. (8.70) has to be extended to

a(M) as L1S=ar- E+ aT L1T.

At constant entropy the temperature variation is given by

L1 T= - [a(M) /~] E. aT aT

Simultaneously, the dipole moment changes as

a(M) L1M=VXTE+ar- L1T.

(8.71)

(8.72)

(8.73)

From the last two relations, the well-known formula for the adiabatic susceptibility follows:

1 [a(M)]2/as Xs= XT- V ar- aT· (8.74)


The various derivatives with respect to temperature are explicitly written as

and

as fJ" 2 aT = T2 L. Pn(hwn-<H») ,

n

a<M) fJ . ar-= T L: Pn(hwn-<H»)(Mnn-<M»). n

(S.75)

(S.76)

The static thermodynamic susceptibilities in (S.67) and (S.74) may be compared with the static limit of the dynamical susceptibility, given by the Green function expression (S.2S), in the case q = 0, w = 0:

(S.77)

It should be noted that all three expressions coincide if the diagonal elements of the dipole moment are zero and no degeneracy exists. If there are nonvanishing diagonal elements or degenerate states, the problem arises how or whether the static thermodynamic susceptibilities can be calculated from the dynamical linear response theory.

The relation of the static limit of the dynamical susceptibility to the isothermal and adiabatic susceptibilities, as derived from the free energy, has been studied by several authors (BRENIG, 1967; WILCOX, 1965; MORITA, 1969; KWOK, 1969; SAUERMANN, 1970). It seems that further physical insight can be obtained from a detailed consideration of specific systems. In doing this it can be demonstrated by examples that response functions in the limit w ~ 0 may not be independent of the limit q ~ O. This has been noticed by COWLEY (1967) in the case of the elastic constants. In the quasi-static limit, the value of the susceptibility is generally not unique but depends on the velocity of the external field. If this velocity is sufficiently low, it is expected that

lim lim X(q, W)=XT' (w/q~O), (S.7Sa) q-+O co_O

whereas for high velocity

lim lim X(q, w) = XS' (w/q ~ (0) (S.7Sb) 0>-+0 q-+O

provided the internal interactions allow local equilibrium to be established at low frequencies by relaxation processes. In the first case, the velocity of the external field is, in the limit w/q ~O, smaller than any velocity of heat transport, so that no local changes in temperature can arise during wave propagation. This is the isothermal limit. In the second case, the velocity of the external field is chosen large, so that no heat conduction can be produced by local temperature gradients. This is the adiabatic limit. Accurate calculations of the susceptibility of dielectric crystals in the limits of(S.78 a) and (S.7Sb) require detailed applications of phonon transport theory. On the basis of the phonon Boltzmann equation, which may be solved in collision time approximations, one can treat the dielectric susceptibility in the same manner as the elastic constants and elastooptic constants have been treated by WEHNER and KLEIN (1971, 1972).


The difference between the isothermal susceptibility XT (8.67), and the static limit of the Kubo susceptibility (8.77), has been discussed recently by PIRC and DICK (1974, 1975). Explicitly including degenerate states this difference reads: (with XI=X(O, 0), the 'isolated' Kubo susceptibility)

LlX=XT-XI= ~ {I Pn IMnm I2-<M)2} ~o, n. m

(8.79)

where the sum includes all states with wn = W m . The adiabatic susceptibility, XS'

could easily be included in the discussion but is neglected here for simplicity. PIRC and DICK have analyzed the conditions under which (8.79) would lead

to zero for the specific case of a two-level system defined by a permanent electric dipole moment which can have two equilibrium orientations and which is linearly coupled with phonons. They used a representation for Ll X with the time correlation function of the dipole operator

f3 1 t LlX=- lim - S d,<LlM(O) LlM(r), LlM =M -<M).

V t~ eX> t 0 (8.80)

The existence of this integral with the limit zero means that the system is 'ergodic', i.e. <M(t)M(0)--><M)2, as t-->oo (CALLEN, 1967). The ergodicity ofa system would, therefore, ensure the equality of the isothermal with the Kubo 'isolated' response in the stationary case but there exists no general proof of the ergodic theorem for general Hamiltonians. Furthermore, in an actual measurement with a finite measuring time it may well happen that the ergodic limit of the time integral has not been reached. This may be considered as a practical difficulty for systems with extremely long relaxation times, which means that the hypothetically existing but unmeasurable isothermal susceptibility has little physical meaning. The non-ergodic case is of principal interest since the question arises whether or not a non-ergodic Hamiltonian can really exist without violating some general physical principles. PIRC and DICK (1974, 1975) have found that their two-level system with strain coupling to the phonons has a non-ergodic limit Ll X =1= 0 as t --> 00 because of the existence of degenerate states in the system as given in (8.79). The degeneracy can be lifted by a small symmetry-breaking term in the Hamiltonian which leads to Ll X = 0, i.e. restores ergodic behavior to the system. On the other hand, the symmetry breaking causes a b-type singularity in the real part of the susceptibility at W = 0 which corresponds to a static permanent moment of the system not related to the Debye relaxation of the original system at low frequencies. This radical change of the system lead Pirc and Dick to the conclusion that adding a symmetry breaking term to the Hamiltonian cannot be used as a general way of making a system ergodic.

One might ask whether this argument is conclusive. A first point to note is the effect of global symmetry on the problem, which has not been considered so far. The relativistic CTP-invariance ensures invariance of a system under C-, T- and, particularly, under P-transformation separately, as long as weak interactions may be neglected. Since the parity operator P commutes with the complete Hamiltonian H, the eigenstates may be chosen to be simultaneously


diagonal for both operators. On the other hand, P does not commute with the dipole operator M, which means that in the stationary (i.e. long-time limit) case no diagonal elements of M exist

Mnn=O, or, <M)=O. (8.81 )

This reduces LIM to M with off-diagonal elements only. It corresponds to the fact that a retarded Green function has only off-diagonal elements of the observable under consideration. The causal Green function which includes diagonal elements is not related to a reponse of the system but rather to the (in this case: constant) propagation of a state or an excitation of the system without an external field. This does, of course, not exclude the approximate appearance of diagonal terms in the analysis of practical measurements if, as mentioned above, the relaxation time for the change of the orientation of the dipole is of 'astronomical' size. This holds, for example, for all experiments where a molecule with an intrinsic dipole moment is assumed to be in a fixed orientation. For this frozen-in configuration the dipole moment must in fact be described by a permanent static, not response-like, correlation function. In terms of eigenstates of the parity operator, <M) may, in this case, be understood as a transition element between an even and an odd parity state which is almost equivalent to a diagonal element of a very long living initial state but not, in a strict sense, a stationary state of a free molecule or crystal in thermal equilibrium.

We are, for the stationary case, still left with the off-diagonal elements of M. On the other hand, the vanishing of the time average <M) implies that a mechanism exists which switches the permanent intrinsic dipole moment, if it points at a certain time into a given direction, after a sufficiently long time into the opposite direction. It is sufficient to know that this mechanism (without knowing any details about it) necessarily breaks the symmetry of the system and lifts the previously assumed degeneracy of the states.

As a conclusion from the foregoing discussion one may conjecture that the equality of isothermal, adiabatic and Kubo susceptibilities may well hold in those cases where the observation times are long enough to arrive at the ergodic limit of the system. In this context, a non-ergodic Hamiltonian should not be used in connection with the 'isothermal' susceptibility of the system. On the other hand, in many practical cases, the relaxation times are much larger than the measuring time, in particular if specific degrees of freedom of the syst~ are practically frozen-in. Then the complete isothermal susceptibility may be of little value and an individual examination of the situation is required. In particular, if a crystal is investigated in a fixed scattering geometry (which is the usual experimental arrangement) its possibly fixed permanent dipole moment (e.g. in ferroelectrics) behaves like a frozen-in dipole transition rather than a dielectric susceptibility.

We have seen before that a high velocity, w/q, of driving the system to the static limit can lead to non-ergodic effects. For the understanding of the detailed behaviour of a system at very low frequencies it is of great importance to analyze the degree of symmetry in the system, for example, inversion symmetry, internal parameters, etc. which allows the classification of crystals into para-


electric, piezoelectric and pyroelectric crystals. The dielectric properties of these different types of crystals have recently been discussed in detail by COOMBS and COWLEY (1973a, 1973b). They show that in non-piezoelectric crystals (i.e. those with inversion symmetry) the susceptibility is equivalent to that given by an ensemble of classically damped oscillators. In piezoelectric but nonpyroelectric crystals (such as the tetrahedrally coordinated semiconductors GaAs, etc.) there is a coupling to the fluctuations in the phonon density distribution away from the local thermodynamical equilibrium, which leads to additional losses at frequencies less than the inverse average phonon lifetime. In pyroelectric crystals, i.e. in crystals where the ions are not all at special positions within the unit cell, there exists a spontaneous polarization in addition to the induced polarization, which leads to an additional coupling to the fluctuations in the local thermodynamic equilibrium. Obviously, the second and the third type of insulating crystals require a very careful examination of the response properties in the regime close to the static limit. The reader is refered to the above-mentioned papers by COOMBS and COWLEY (1973a, 1973b).

9. Raman scattering of light

a) Introduction. In Subsect. 7 d we have outlined the general theory of inelastic light scattering by electrons. We are now presenting the theory for the case where the final states in the crystal are such that one or several phonons are excited or de-excited after the scattering process. There are a few points to notice at the beginning of our discussion.

The origin of phonon Raman scattering is a change of the crystal polarizability due to the interaction of photons with electrons which on their part excite phonons via the electron-ion interaction. The frequency of the external light is usually high compared with the phonon frequencies. Therefore, the ions cannot follow the frequency of the external light, i.e. they stay practically at rest while the electrons are polarized by the light. In a rough estimate of the order of magnitude, the Raman cross section for ionic polarization is down by a factor of

as compared to the electronic Raman effect. This is equivalent to saying that the adiabatic approximation is very well fulfilled for off-resonance Raman scattering. We note that the ionic Raman effect has so far not been observed experimentally (refer to: GENZEL and MARTIN, 1974).

The adiabatic condition is violated if the difference between the light frequency and the lowest real excitation energy of the electrons becomes comparable to the phonon frequencies. In this case of resonance Raman scattering the possible influence of non-adiabatic electron-phonon coupling has to be considered (see Sect. 9c).

For the remainder of our discussion we use the adiabatic approximation, i.e. we replace the crystal Hamiltonian H by its expectation value in the electronic ground state, leading to the crystal potential cp, (34.22). In the dipole

Sect. 9 Raman scattering of light 109

approximation for the interaction Hamiltonian, Hdip in (7.94), the inelastic light scattering was described in terms of transition-polarizability amplitudes P(wiws)' (7.76). These consist of matrix elements of the dipole operator m (7.95), which depend on the positions fL of the ions. They can, therefore, be expanded into displacements of the ions from their equilibrium positions. This treatment leads to the polarizability theory of Raman scattering which is called the (generalized) Placzek approximation (PLACZEK, 1933; BIRMAN, 1974) and is summarized in Sect. 9b and in Sect. 13 after a discussion of the lattice potential.

An alternative approach emphasizes the analogy of Raman scattering with infrared absorption as a response mechanism and uses the formalism of retarded Green functions for the calculation of the Raman cross section (Sect. 9 e). This approach allows a systematic treatment of Raman scattering as a many-body problem and may therefore be useful for further developments of the theory.

Finally we discuss (Sect. 9g) the polariton picture of Raman scattering which is an extension of the harmonic polariton treatment in Sect. 7f. The starting point of this method is the fact that photons propagating in a crystal with a frequency far above the phonon frequencies but still distinctly below the electronic energy gap may be described as quasi-particles with a well-defined self-energy. The real part of this self-energy describes the reduction of the light velocity by a factor equal to the refractive index while its imaginary part corresponds to an exponential damping of the photon amplitude. This picture, while quite useful in the description of photon-phonon and photon-electron interaction at long wavelengths as mixed polariton excitations including their dependence on the wave vector, does not seem to be not very helpful for an interpretation of second- or higher-order quasi-particle excitations in a crystal. The main reason for this objection is - as mentioned in the discussion of infrared absorption - that these excitations involve preferentially phonons (or excitons) of short wavelengths for which the deviation of the photon wave vector from zero is usually irrelevant. Anyway, the polariton concept considers only an intermediate state in the crystal while the experimental scattering cross section gives the ratio of the incident to the scattered free photon currents outside of the crystal and does not involve explicitly any polariton property.

b) Quantum theory of spontaneous Raman scattering

Q() Free particle Hamiltonian and interactions. In Sect. 7 e the light scattering by electrons was characterized by a second-order matrix element, the transition polarizability amplitude P. We describe the scattering process here in somewhat more detail and specialize the discussion to the case where the electrons take part in the process only by intermediate transitions from the electronic ground state to higher states while phonons are excited or destroyed in the crystal as a result of inelastic light scattering. The complete system of radiation and matter is described by the Hamiltonian, (7.80), (7.88),

(9.1)


with the radiation part Hr' (721), the crystal Hamiltonian H, (7.89), and finally the interaction Hamiltonian Hint' (7.90) and (7.91), which we shall use in the dipole gauge, (7.41)-(7.43), and in the dipole approximation, Eq. (7.37).

{3) Static representation. The electron-ion interaction is often presented in the so-called static representation i.e. all electronic wavefunctions are determined at the equilibrium configuration, Ro, of the lattice ions. This allows the use of wave functions or matrix elements already known from a tightbinding or pseudo-potential calculation of energy bands or electronic optical absorption. In addition, results of resonant Raman scattering (see Sect. 9 g) are straightforwardly related to band structure quantities while adiabatic effects due to the displacement of ions are considered by renormalization of electronic states. For a recent discussion refer to ALLEN (1981).

Here we begin with a discussion of Raman scattering in terms of the static representation while the adiabatic one (which is closer to the general concepts of lattice dynamics) is discussed below (Sect. 9c). We assume that the scattering process can be described by transitions from well-defined initial and final states of the system which are eigenstates In) of the isolated crystal Hamiltonian, H, (7.89), i.e. joined phonon-electron states. For the purpose of the following discussions we introduce a more detailed description by defining a 'free particle' part of the Hamiltonian Ho, while the remainder V defines the different types of interactions in the system:

Htot=Ho+ V. (9.2) Here

H 0= H r + He + H ph' (9.3) where

HrIN!') = n(N!, + t)W!, IN!') (9.4)

defines the free photon states IN!'), and

(9.5)

corresponds to the electronic energy in the static approximation while

(9.6)

describes the phonons (with quantum numbers A=(q,j)) in the harmonic approximation.

The interaction part is then:

V =He.r +He •ph + H ph •ph

with

H e.ph = <Pe. ph

and

Eq. (7.94),

(see below)

Eq. (34.23), n ~ 3.

(9.7)

(9.8)

(9.9)


In the new representation the crystal Hamiltonian is given by

H =He + Hph +cPanh +cPe,Ph' (9.10)

The theory of photon scattering by charged particles has been given in Sect. 7. The cross section for Raman scattering was described in terms of an effective interaction potential, veff, (7.9), which contains the photon-electron interaction H dip as well as the electron-phonon interaction cPe,ph' The final determination of the cross section needs the evaluation of veff. This operator determines the order of perturbation theory which is given by the number of intermediate states required for a transition from 10) to If) with the interaction potential 11; (9.7). The only part of V which leads to a creation or destruction of a photon is H dip' Since it is linear in the electric field we need H dip in second order for an inelastic scattering process where a photon with energy nWi is destroyed while another photon with energy nws is created. The dipole operator m, (7.95), is linear in the electronic and ionic coordinates r, and rL and can therefore, for its part, in a second-order process, lead to elastic light scattering. Since real transitions of electrons are not considered, this is the only possibility of scattering in second-order which determines the electronic polarizability Boo of the crystal in the optic frequency regime.

Raman scattering is therefore at least a third-order process, if one-phonon processes are considered but of fourth- or higher-order for two-or manyphonon processes.

The general structure of third-order perturbation theory (refer to MARCH et aI., 1967) gives in our case, with the notation of Sect. 7b{J,

V;:ff = I <il V 1m) <ml V 1m') <m'l V Is). m,m' (Em-EJ(Em,-EJ

A typical term of (9.11) is

<nl H dip 1m) <ml He,Ph 1m') <m'l H dip In) (Em - En - nwJ (Em' - En - nWi + nQ)

(2nnwy/2. dnm · Dmm,· d!'n' (2nnw//2

(Wi + Wnm - iB) (Wj + Wnm' - iB)

with the dipole matrix elements, (7.54),

ednm = <nl m(r)· e(q) 1m) = eRe(q)

(9.11)

(9.12)

(9.13)

(9.14)

calculated with equilibrium electronic wave functions and electron-phonon matrix elements

Dmm, = <ml He,ph 1m'), (9.15)

the explicit form of which depends on the specific approximation used in the calculation.


The cross section now becomes (refer to (7.13))

(9.16)

with the fourth-rank tensor

(9.17)

(9.18)

Note that the corrections due to refractive indices n have been omitted.

c) Adiabatic representation. In Sect. 3 d the adiabatic approximation was discussed. It allows a separation of a lattice potential cP from the remainder of the crystal Hamiltonian H, (7.89). The harmonic part of cP then defines the harmonic phonons.

To give more detail let us proceed as follows (refer to BORN and HUANG, 1954). The lattice potential is defined by the expectation value of H - r; == H ei in the many-electron ground state 10) which depends only parametrically on the displacements u(L) of the ions of the crystal from the equilibrium configuration,

(9.19)

Here we split the total electronic system into a set of valence electrons which appear explicitly in the theory and a set of core electrons that enter through a modification of the ionic potential U;, (7.86), by replacing the bare charges Zn of the nuclei by ionic charges which contain the effect of the core electrons and of the orthogonalization of the wave functions (refer to BILZ et al., 1974).

This implies that we assume that the cores are non-overlapping. The Coulomb interaction between the cores is denoted cPi in (9.19), while the remainder cPe is the adiabatic electronic energy which, in particular, contains the electron-ion interaction, (7.87), but with ionic charges taken as stated above. The adiabatic crystal Hamiltonian is given by

(9.20)

The crystal potential (34.22) may be expanded in a power series in the ionic displacements u(L), (34.23). cP2 defines the harmonic potential for the determination of phonons, cf. (3.16) and (34.5). (Note the slightly different notations.) It can be expressed by formal two-body potentials;

cP2 =1 L cP2(L, L) (9.21) L,U

with the electron-ion part (cf. (7.87))


tPe.2(L, L)= aU~L) <OlaU~E) U.i 10) u(L) u(E), (9.22)

where the so-called Hellmann-Feynman theorem has been used. Equation (9.22) can be regarded as the starting point for all microscopic theories of phonons in the adiabatic and harmonic approximation.

The difference between the crystal Hamiltonian and the lattice potential defines the non-adiabatic electron-phonon interaction for the ground state 10)

(9.23)

It plays no role in lower-order infrared absorption processes since in this energy regime the electrons always remain in their adiabatic ground state. Transitions between different phonon states are then caused only by the photon-phonon interaction Hint ';;;; Hdip and the phonon-phonon interaction

(9.24)

In Raman scattering i.e. in the frequency regime of eV the ions cannot follow the electrons so that the electronic energies are calculated for one fixed configuration of the ions, namely the equilibrium configuration. The electronic wave functions of this static approximation are, however, not immediately useful for the determination of the lattice potential 4>, (9.19). The difference between the adiabatic and the static potential (i.e. calculated with the static wavefunctions) for the electronic ground state is given by

with

4>0ad_4>0st= (0, RI Hei 10, R) -(0, Rol Hei IRo, 0)

= <0, RI Uei IR, 0) - <0, Rol Ue~ IRo, 0)

(ORI Uei 10R)R~Ro=<ORol Ue~ IRoO).

(9.25)

Here, R o and R are short notations for the ionic configurations in equilibrium and displaced from it.

The difference between 4>ad and 4>st may be described in terms of the perturbation treatment of BORN and OPPENHEIMER (1927) who showed that the parameter K=(mjM)1/4 is the appropriate expansion parameter in the adiabatic approximation, where m and M denote electronic and ionic masses, respectively. This means that we are still in the framework of the adiabatic approximation if n in (9.24) is not greater than 4 (cf. BORN and HUANG, 1954; ZYBELL, 1972). An investigation of (9.25) leads to the result that 4>ad - 4>st is of the order K2 ie. of the same order as the harmonic approximation and is therefore substantial in a calculation of phonons. (For a recent discussion see GRIMV ALL (1981).)

Since in Raman scattering theory the electronic wave functions are usually assumed to be those of the static approximation, i.e. determined at the equilibrium configuration, we have to consider an explicit correction for the difference


between the static and the adiabatic lattice potential of the phonons. We then obtain for the difference between the crystal Hamiltonian H and the Hamiltonian for the decoupled system of electrons (static approximation) and phonons (adiabatic approximation), instead of (9.25),

(9.26)

with the adiabatic electron-phonon coupling

(9.27)

Since this term is of order ,,2 but the non-adiabatic term in (9.26) begins with a contribution of order ,,5 (refer to BORN-HUANG, 1954), we may expect that the adiabatic electron-phonon coupling usually dominates the situation while the remainder may be neglected.

This discussion shows that infrared absorption and Raman scattering, at least for low-order processes, are determined by quite different non-linear couplings. The first (low-energy) process is determined by the anharmonic part of the adiabatic lattice potential while Raman scattering involves even lowerorder ("harmonic") parts of the electron-ion interaction. Particular care is required for the model theory (Sect. 14), otherwise a mixing of 'anharmonic' and 'non-linear' electron-ion couplings is almost unavoidable.

For a systematic treatment of electron-ion coupling in the static approximation we refer to ALLEN (1981).

oc) Unitary transformation. An elegant and powerful way to investigate electron-phonon interactions is that of unitary transformations. While in standard textbooks electron-phonon interaction in insulators is usually described by an expansion technique (e.g. MADELUNG, 1978) one finds occasionally the application of unitary transformations to specific problems such as the polaron problem (refer to DEVREESE, 1972; NAKASIMA et at, 1978) For metals a discussion of the method has been given by CHESTER (1961). A very recent investigation of unitary transformations in solid state physics can be found in a monography by WAGNER (1983). Here, we follow a paper by ZYBELL (1972) since he approached the problem in close analogy to the model theories of lattice dynamics.

For the calculation of matrix elements of He. ph we have to consider that the essential part of He,Ph is in a mixed representation, (9.27),

(U:~h)nm = <n, RI L1 Uei IR, m) - <n, ROI L1 Uei 1m, RO).

We may use a unitary transformation which transforms the displaced configuration, {R}, into the equilibrium configuration, {RO}, by (ZYBELL, 1972):

(9.28)

This yields, in a short notation,

Uad =<RoID IRO) e,ph R' (9.29)


with (9.30)

The unitary transformation TR may be determined in a certain approximation for the calculation of one- or multi-phonon transitions. The procedure has something in common with the method of GANGULY and BIRMAN (1972) but uses dipole matrix elements instead of those of the momentum operator. The latter ones are not gauge-invariant but have the advantage to be more easy to calculate. We come back to this point later in Sect. 14b.

d) Polarizability theory (PLACZEK, 1934). To make contact with the usual representation of the scattering tensor i in terms of transition polarizabilities P we go back from the representation of the interaction V in terms of matrix elements with static electronic wave functions to those with adiabatic wavefunctions. Then, the dipole matrix elements are displacement-dependent and, instead of using explicitly the electron-phonon interaction He. ph we may use an expansion of the P's with respect to ionic displacements. The transition rate now becomes implicitly dependent on the ionic displacements

(9.31)

The transition polarizability is given in the dipole approximation by (7.76); for the purpose of this section it may be transformed with the help of a Fourier transform of the dipole operator, (8.6), into the form

~nr(qi' q.)= S d3 rei(Qi-q,jr ~p(r)

=~ L [<nl ma~.) Iml><m/l~!(q) In> h m' Wm'n+Wi 1£

(9.32)

<nl ma(q)lml><m/l~p(q.) In>]. w m •n +Ws -l£

This shows clearly the form of a second-order perturbation theory with displacement-dependent dipole moments m. Equivalently, we can replace the secondorder perturbation theory with the dipole interaction by a first-order theory with an effective Hamiltonian:

veff = -i S d3 rE(r) bx(r) E(r) (9.33) with the susceptibility

b x(r) = 8P(r)/ 8E(r). (9.34)

Considering the finite temperature of the crystal we use the density matrix to calculate proper thermal averages

(9.35)

with p-l=kT, and w;. denoting the different phonon energies. We obtain then the scattering tensor


n.m

(9.37)

where the brackets < ... ) indicate a thermal average. By comparison with Sect. 7 a we see that the Raman scattering tensor is essentially the Raman dynamic form factor S( Q, Q).

The transition polarizability P in (9.37) is related to the electronic susceptibility in a simple manner:

L Pn~pn(q, OJ) = X~fJ(q, OJ) (9.38) n

or <P(q, OJ) = l(q, OJ). (9.39)

Since we are interested in the effect of phonons, we have used Placzek's approximation (cf. BORN and HUANG, 1954; PLACZEK, 1934), i.e. assumed that the electrons are in the ground state at the beginning and at the end of the interaction, so that the quantum numbers nand m merely denote the phonon states. The polarizability will therefore be expanded into phonon normal coordinates in Sect. 13 so that explicit formulae are obtained for one-phonon and multi-phonon scattering processes in·crystals in various approximations.

e) Green function theory of Raman scattering (WEHNER and KLEIN, 1972). An alternative approach proceeds in an analogous way as for infrared absorption in Sect. 8. There, we described the absorption as the imaginary part of a linear response function which was given by a retarded Green function or, equivalently, as the spectral density function of the fluctuations of the dipole moment. In an analogous way we can develop the theory here by replacing the dipole moments by the polarizabilities or susceptibilities of the crysta1. The Raman cross section may be interpreted as a non-linear response function due to the fluctuations of the electronic polarizability. These fluctuations scatter the incident photon and may therefore be used to define a photon self-energy of which the imaginary part is the finite photon lifetime due to Raman scattering.

We use the general results of Sect. 35 (refer to WEHNER and KLEIN, 1972) and begin with the retarded Green function of an incident photon with quantum numbers Pi=(qi,Si) and Pi=(-qi,Si) is defined by

(9.40)

Here, normal coordinates A(p) are introduced and the vector potential A(r), the electric field E(r) and the magnetic field B=curlA are written in analogy to phonon field operators, A(A), Sect. 34, with (J = - 1,

(9.41) p

E(r) = L (2nhOJpjV)1/2 e(p) A(p) (OJp/qc)eiqr, (9.42) p

Sect. 9 Raman scattering of light

A(p) =a(p) + at(p)

A(p)= -iwp[a(p)-at(p)].

117

(9.43)

(9.44)

The a, at are phonon destruction and creation operators with the usual commutation rules (cf. (34.19)) for the analogous case of phonons; c is the velocity of light in vacuum. The phenomenological Hamiltonian of the radiation field in the nearly transparent frequency regime of the crystal has the form

1 Hr=~S d3 r[I s~pE~(r)EfJ(r)+B2(r)]

8n ~,p (9.45)

= h I wp[at(p) a(p) +tJ, (9.46) p

where 8~,p is the frequency-dependent dielectric tensor of the crystal in the equilibrium configuration. The polarization vectors obey the condition e(p) = e*(p), and the light frequency is determined by

w~ =(cq)2(I e:(p) 8~p(Wp) ep(p))-l. (9.47) ~.p

If the nuclei are allowed to vibrate, their motion is represented by the lattice Hamiltonian which includes Coulomb forces but no retardation. The effect of the phonons on the light propagation is then taken into account by introducing a space- and time-dependent electronic susceptibility b x(r, t). It is assumed that the fluctuations depend only on the displacements of the atoms from their equilibrium positions so that they can be expanded in terms of phonon normal coordinates. Microscopically this implies a coupling between the phonons and the electrons, since virtual transitions of the latter are the basic mechanism of the coupling of light to the crystal. The fluctuation is related to a change in the energy of the photon-phonon system giving rise to the interaction Hamiltonian, (9.33),

Here, the coefficient (w 1 W 2?/2

bXP1P2 =2n I 2 e~(pl) ep(P2) bx~p( - ql- Q2)' ~,p c ql q2

(9.48)

(9.49)

(9.50)

is expressed by the Fourier transform of the spatially varying fluctuation bX. Then, the total Hamiltonian is

(9.51)

This Hamiltonian describes the photon-phonon system in an approximation suited to treat scattering of light at optical frequencies which are high compared to the phonon frequencies. Not included are the photon-phonon interactions via the ionic charges which are responsible for infrared absorption and also account


for scattering processes with intermediate states reached by virtual phonon transitions. As stated before, the latter processes are not important as long as the light frequency is much higher than the phonon resonances.

With bXP1P2 expanded into phonon normal coordinates, the Hamiltonian of (9.48) is identical in form to the general anharmonic phonon Hamiltonian; this means that the photon-phonon system can be considered and treated like the phonon system itself, but with two additional branches representing the transverse radiation field. In the following, this is done by applying the general methods and special results from anharmonic phonon theory.

For the determination of the transition rate of light scattering processes the self-energy of the incident photon must be analysed. This is done by obtaining the equation of motion of the retarded Green function (see Sect. 36), given in (9.40). With the help of the equations of motion for the normal coordinates (9.44) and

p,

we obtain for the Green function the following equation

{:t22 +W;i} Gt_t,(A(pJ I A(o;))=2wpi b(t-t')

+ 2WPi L Gt_t,(bXPiP,A(ps) I A (,0;)). Ps

(9.52)

(9.53)

When the new Green functions on the r.h.s. are differentiated with respect to t' the result is

{a~~2 +W;i} Gt_t,(bXPiP,A(ps) I A(P;))

= 2WPi L Gt-t,(bXPiPS A(Ps) I bXPiP,' A(ps))' (9.54) Ps'

because bX and A(p;) commute for equal time. Taking the Fourier transform of (9.53) and (9.54) results in

Gro(A(pJ I A (,oJ)

= Gp(Oi)(W) + Gp(O,)(w) '\' G (b A() I "' A()) G(O)( ) • L., ro XPiPs Ps uXPiPs' Ps Pi W, (9.55) PsPs'

where (O)()_ 2WPi

G Pi W - 2 ( ')2 ' WPi - W+1(; (9.56)

(see (34.50)). The simplest approximation for the irreducible self-energy of the photon p; is

given by

Gro(bXPiPsA(ps) I bXPiPS,A(ps)) G~~)(w)

~ G~(bXPiP,A(p.) I bXPiP, A (,os)) bp,ps' Gro(A(p;) I A (,0;)). (9.57)

Here, the Green function GH is understood to be in the Hartree approximation where the corresponding correlation functions factorize:

J~(bXPiP' A(ps) I bXPiPsA(,os)) = Jt(bXPiP, I bXPiPJ Jt(A(ps) I A (,os))' (9.58)

Sect. 9 Raman scattering of light

From (9.57) and (9.58) the self-energy follows as

IIPi (w) = LI Pi(W) - i r;,i(W)

= - L G~(i5XpiP,A(ps) I i5XPiP, A (Ps))' p,

119

(9.59)

The total inverse photon lifetime ~ = 2 r;,i (w;) is the sum over partial inverse lifetimes r Pi

1 H _ -r -. = 2 1m G Wi (15 XPiP, A(ps) 115 XPiP, A (p,))

PiPs

1 1 + n(wJ J:!, (15 Xp,p, A(ps) I 15 Xp,p, A (Ps)), (9.60)

which are calculated by taking the Fourier transform of (9.58)

J:!Ji5XPiP, A(ps) I i5XPiP, A(ps)) 1 00

= 2n S du J Wi _u(i5 XPiP,! 15 XPiP,)Ju(A(p,) I A (Ps))' (9.61) -00

For weakly absorbing materials, the spectral density function of the photon Ps is of the form

Ju(A(ps) I A(p,))=2n{(1 +ns) i5(u-ws)+ns i5(u+ws)}, (9.62)

which in (9.60) leads to

_1_ = 1 + ns J. (15 _ I 15 _ ) + ~ J. (15 _ I 15 _ . 1 + . W,-w, Xp,p, XP,Ps 1 + . w,+ws Xp,p, Xp,p,)

~~ ~ ~ (9.63)

Disregarding stimulated scattering, ni and ns are equal to zero. In this limit, (9.63) can be interpreted as the rate of scattering of the photon Pi into the mode Ps' Introducing the energy transfer Q=wi-ws, this rate is finally written as

(9.64)

The scattering rate per solid angle element, dQs' and per energy interval dws is obtained using (7.5)-(7.8)

(9.65)

With dqs=dws·e1/ 2/c, and w;=(cq)2/e this gives the spectral differential scattering cross section (Q = q i - qs)


This result expresses the scattering rate (9.65) in terms of the susceptibility autocorrelation function commonly used in the calculation of spectra, and is similar to (9.16) and (9.36). It must be emphasized, however, that dQs denotes a solid angle of radiation inside the crystal, whereas the intensity of the free, scattered radiation, as measured outside the crystal, is more conveniently defined with respect to a different solid angle element, dQs (cf. Fig. 7.1), which is related to dQs by the surface geometry and by the laws of refraction. Assuming, for simplitity, that the scattered ray passes the surface vertically, we have dQs = 6 1/2 dQs, so that the relative intensity of the scattered light per unit solid angle in free space and per unit frequency interval can be written as

(9.67)

Here the Raman scattering tensor i is given by

To make the formal similarity of infrared and Raman processes more apparent, we may use the relations between correlation and Green functions as discussed in Sect. 34. It is sufficient for the purpose of this section to use the dynamical part J~ of the spectral density function JQ (see (34.67)). Then, with (34.75) and (34.59), one obtains

JQ(bX I 15 X) = 2 [n(Q) + 1] 1m GQ(b X 115 X)

J _Q= [n (Q)j(n (Q) +1)] JQ ,

(9.69)

(9.70)

where JQ are the Stokes and J_ Q the anti-Stokes components of the Raman scattering and GQ(bxlbx) the retarded Green function of the susceptibility bX in the (w, q) representation. Therefore, the Raman tensor for the Stokes component is

(9.71)

A comparison of this equation with that for the susceptibility X~fJ in (8.24) shows their formal similarity.

f) The w4 -law. In the foregoing sections we have seen that the differential Raman cross section obtained by different methods is of the form

d2 (J" 3 • dQs dws ocwj Ws . l, (9.72)

where the elements of the Raman scattering tensor i are built up from the matrix elements of the susceptibility fluctuations

i~fJ'~'fJ,ocbX~fJ· bX~'fJ'· (9.73)

The explicit frequency dependence of the cross section on Wj w: reflects the kine tical part of the scattering process which is given by the incoming and outgoing photon flux (OCWj and Ws respectively) and by the phase space which gives a factor w; due to the fact that every scattering angle contributes with a differential phase volume element q; dqsocw; dws. This means that the Raman


cross section would exhibit an w4 law only if the susceptibility bl does not depend on the external light frequency Wi~ws. We observe this behavior in the off-resonance regime of insulators. It does not hold, however, in the case of resonant Raman scattering where the frequency dependence is determined by certain interband resonances. A further case is the scattering by nearly free electrons where bXcx.W- 2 holds, so that the whole cross section becomes nearly constant due to a compensation of the kinetical by the dynamical part of the cross section. A source of confusion seems to be the fact that in a p. A treatment (instead of the r· E representation for the dipole interaction) the explicit Wi w; dependence of the cross section is easily masked by an implicit frequency dependence of the dipole matrix elements (LOUDON, 1963). The problem is analogous to the discussion of the difference between the p. A and the r· E representation for the dielectric susceptibility, Sect. 7 d. There, we have seen that every matrix element of the r-operator corresponds formally to one of the p-operator divided by the transition frequency (refer to (7.62}):

(9.74)

In order to see that the replacement of matrix elements rOn by POn in practice destroys the explicit frequency dependence of the Raman cross section, we shall now discuss the case in somewhat more detail since the use of matrix elements POn of the momentum operator is a practical necessity in calculations based on energy-band structure methods. We follow here a recent discussion by ZEYHER et al. (1976).

We start by mentioning that the various methods which have been used to derive Raman cross sections seem to agree with each other if the same Hamiltonian and the same notation is used (BARKER and LoUDON, 1972). In the case of molecules both the p. A and the r· E coupling have been used (the A 2 term does not contribute in the dipole approximation). The first one leads to an explicit ws/Wi factor in the cross section whereas the second one leads to a w; Wi factor well known from the semi-classical treatment. The equivalence of both approaches within the dipole approximation has been proven by BA YM

(1969). In the case of infinitely large crystals only the p. A coupling leads to well-defined matrix elements. Correspondingly, all the cross sections referring to crystals exhibit only an explicit Ws/Wi instead of the W;Wi factor. We are now going to show that the commonly used formulae for cross sections of crystals can be recast in a form in which an explicit W~ Wi dependence appears.

The differential cross section for inelastic light scattering (9.32), using the p. A coupling can be written (refer to BA YM, 1969)

(9.75)

ei and eS denote the polarization of the incident and scattered light, respectively, and f3 = 1/kT. ROn(w + ill} is the equivalent of pOn, (9.32), which was given in terms of matrix elements er~n:


<01 Rp 1m.> <ml Ra In)] . W s +1i;+WmO

(9.76)

10), In) and 1m) are exact many-body eigenstates of the crystal without the radiation field and denote the initial, final and intermediate states with the energies hwo, hWn and hwm . Ra is the momentum operator of the crystal (refer to (7.83), (7.84)):

n = L (- PI) + L (Zn ~), I m n Mn

(9.77)

where the indices I and n count the electrons and cores, respectively. The r· E coupling has in our case the form (refer to (7.94))

HdiP= -M·E (9.78)

with the total dipole operator (refer to (7.95))

M = I ( - e r1) + L e Z n rn' (9.79) I

The equivalence of the n . A with the M· E coupling means

e - <min· A In) ~ <ml M· E In) me

(9.80)

or, with (9.74) and E= -i(w/c)A, Rnm equivalent with (W/Wnm)Rnm in the Raman scattering tensor. We may therefore replace R, (9.76), by the equivalent Raman tensor R:

<01 Rp 1m.> <ml Ra In)]. (9.81) Ws+le+Wmo

The above argument shows that the Raman tensor can always be written in the form of (9.81). For actual calculations this equation has many advantages compared to (9.76). It always leads to the observed W: Wi dependence in transparent frequency regions independent of the number of states taken into account in the intermediate sum. On the other hand, (9.81) needs an extremely careful evaluation of the intermediate sum to yield meaningful results outside the resonance region and to reproduce at least approximately the W: Wi dependence. Connected with this is also the fact that the intermediate sum in (9.81) has much faster convergence than (9.76) due to the additional factors wjwmo and ws/wmn •

Explicit formulae for Rand R describing inelastic processes such as scattering with phonons are obtained from (9.76 and 9.81) by approximating the exact crystal states in an appropriate way. In the case of phonon Raman scattering, these states are calculated using perturbation theory up to a certain order in the electron-phonon coupling and all the resulting contributions to R or R up to the same order are kept. One finds in this way for R six different terms for first -order and the different terms for second-order Raman scattering given in Tables 40.5-40.7. The corresponding contributions to R are obtained by applying the same procedure to (9.81). The general method to calculate Raman tensors in the form of (9.81) leads to an explicit W~ Wi dependence for Raman scattering.


We note finally a very simple connection relating the contributions of each intermediate state to the symmetric part of the Raman tensors, Rand ii.. One easily verifies the following identity

Taking matrix elements into account, one obtains

Rm=Rm+c~n,

(9.82)

(9.83)

where m denotes the contribution of the m-th intermediate state to R, Rand C. C is a constant independent of the frequency Wi' For Wi = 0, Rm vanishes so that (9.93) can also be written as

R(wi)=R(wJ-R(O). (9.84)

If Rm is represented by a power series in the electron-phonon coupling, (9.93) holds for each order separately. This means for instance that (9.93) holds for the Raman tensor for first-order scattering if all the six terms are taken into account. Since the identity (9.91) is true for each state m, (9.94) can also be used for a sum over a finite number of intermediate states m. It seems, therefore, that (9.93) offers the most convenient way to transform the symmetric part of the Raman tensor from the form R to the form R.

Relation (9.94) is not true in general for the antisymmetric part of the Raman tensor. On the other hand, the anti symmetric part is non-zero only for resonant scattering for which the expression (9.94) is already acceptable.

We note again that this discussion does not hold for quasi-free electrons (such as conduction electrons in a doped semi-conductor) where the Raman polarizability is proportional to W - 2.

g) Polariton picture of light scattering. In Sect. 7 f we have described the interaction between radiation and matter in the harmonic and dipole approximation which leads to new quasi-particles of the combined system, i.e. polaritons. The Hamiltonian of the harmonic polaritons can be derived by a unitary transformation of (9.5) and (9.7) (cf. (9.46))

Hr + Hph ~Hpol = h L OJ;. [txt (A,) tx(A) +tJ, (9.85) ;.

where the txt (A), tx(A) are creation and destruction operators for polaritons, the frequencies of which, OJ;., are solutions of (7.103).

In the general case we have to consider the anharmonic part of the lattice potential and the non-linear electron-ion coupling. As indicated in the foregoing section in a phenomenological manner, we may try to introduce complex selfenergies for the polaritons in such a way that the lifetimes of these excitations describe correctly the details of light absorption and scattering.

This picture is useful for the analysis of one-polariton spectra but it looses its advantages when going to multi-polariton spectra. It seems then, that an explicit treatment of the interactions between the radiation field and the different quasi-


particles in the crystal is clearer and thus preferable, in particular if boundary conditions at the surface of the crystal have to be considered.

An exact Hamiltonian has been given by MAVROJANNIS (1968) for the rigidion model using the polariton concept. In that article the approximation of neglecting the magnetic-field component in the interaction means that in the exact treatment of the rigid-ion model the gradients of the electromagnetic vector potential are neglected, i.e. the vector potential at the actual position of the displaced ion is taken to be that at the equilibrium position. This approximation is irrelevant for the definition of the polaritons, as long as they are derived from the second-order terms of the photon-phonon interaction. However, as a consequence of this approximation all interaction terms which in the rigid-ion model lead to the second-order dipole moments and to light scattering vanish. This is a well-accepted approximation for the rigid-ion model. The physically important mechanisms in a dielectric crystal for light scattering and for infrared absorption by non-linear dipole moments are not the gradients of the vector potential but are virtual electronic transitions, driven by the electric field. Through electron-phonon coupling, these transitions lead to photonphonon interactions formally described by the expansions of the dipole moment and the susceptibility into phonon coordinates. Recently, MAVROJANNIS (1970, 1973) has extended his treatment of polaritons.

h) Resonant Raman scattering (RRS). In the foregoing section it was assumed that the energies of the incoming or scattered light were always sufficiently below the optical gap of the solids under consideration. This condition allowed us in many cases to neglect the detailed nature of the excited electronic states and to focus attention on the effect of the modulation of the electronic charge density by phonons in a low-frequency external field. This "classical" regime turned out to be very appropriate for the use of model theories which are powerful in making explicit contact between the lattice dynamics of a certain family of solids and their light scattering properties. The excited states in a crystal could be represented by an 'effective' gap of the order of the lowest ionization energy or the corresponding model electron-ion forces.

In the case of semiconductors with gaps between 1 to 3 eV many laser frequencies are close to or even above the lowest band gaps and RRS becomes possible. Here, the specific features of one or few electronic transitions and their particular coupling to the phonons of the system are dominating. Many new and interesting physical aspects come then into play such as the lifetime of intermediate states etc. We are not going to give a detailed account of RRS in this article but concentrate our discussion on the general structure of the electron-phonon coupling in this case. The reader who is interested in a more detailed discussion of RRS is referred to the Proceedings of the three last International Conferences on Light Scattering (WRIGHT, 1968; BALKANSKI, 1971; BALKANSKI et aI., 1975) and to some recent reviews in the field (BIRMAN, 1974; CARDONA, 1975; RICHTER, 1976; GONTHERODT and CARDONA, 1982). A short comprehensive account of RRS has been given by RICHTER and ZEYHER (1976).


In RRS electronic transitions are near or at resonance with incident photons. Often, the scattered spectrum contains also a certain admixture of luminescence. This part of the secondary emission spectrums differs from RRS by the final widths of intermediate states while in RRS line broadening is due to initial and final states only. Experimentally, one sometimes can separate luminescence from RRS by the frequency-dependent and rather strong linewidth of electron-hole excitations as compared to the frequency-independent and small phonon linewidth in RRS.

RRS shows often similarities with other optical experiments. This holds in particular for 'vertical' excitations (qi ~O) and momentum-independent bandenergy differences. In this case optical (transmission or reflection) experiments provide sometimes better information about (weighted) electronic densities of states of interband transitions then RRS.

Most important for the analysis of RRS are one- and two-phonon deformation potentials which describe an essential part of electron-phonon coupling in terms of selected electronic energy band states and related phonons. In this respect, the information obtained from RRS is complementary to the offresonant Raman scattering which preferably provides information about the local (i.e. on-site or near-neighbor) deformation potentials in terms of differential ionic or bond polarizabilities. This aspect of Raman scattering dominates the discussions of forthcoming sections because it is closely related to concepts and quantities of lattice dynamics and infrared absorption.

i) Rayleigh, Brillouin, and Hyper-Raman scattering. The study of very-low frequency excitations in a solid are often done by using Rayleigh and Brillouin scattering. The first technique is useful to investigate quasi-elastic scattering (entropy fluctuations) while the second one gives information about sound waves (cf. SANDERCOCK 1982).

A very interesting experimental method is Hyper-Raman scattering where the non-linear electric dipole moments come into play. This technique provides a promising research area complementary to infrared absorption (cf. VOGT 1982).

D. Expansion theory of susceptibilities and polarizabilities

In this chapter, the dynamical properties of electrons and phonons will be investigated as far as they are important in Raman or infrared processes. It is assumed that the crystal states can be described by a certain net number of phonons, with definite wave vectors q and branch indices j, which have been excited in the crystal (absorption or Stokes scattering) or taken from its phonon reservoir (emission or anti-Stokes scattering). The electrons are always supposed to be again in their initial states after the interaction has taken place, i.e. the dynamical response, of the electrons is restricted to the virtual processes of polarization type. It is well-known from quantum mechanics that such polariza-

126 Expansion theory of susceptibilities and polarizabilities Sect. 10

tion processes may be satisfactorily approximated by a two-level system with an effective gap and an effective oscillator strength. This allows for a pseudoclassical description of the electronic excitations in terms of oscillators (refer to Sect. 4).

An important tool for the understanding of crystal excitations is the study of the change of external parameters such as temperature, pressure, and static fields. Therefore, the discussion of effects induced by a variation of these parameters will be covered in this chapter. Many of the results given in the following sections have been derived by BORN and HUANG (1954).

10. General lattice potential.

a) The undeformed lattice. In Sect. 3 the normal vibrations of the lattice were introduced in the adiabatic and harmonic approximations. Each phonon is characterized by a wave vector q and by a branch index j, and determined by its frequency W;. =. w(q j) and polarization vectors e(4~,). The definition of these vectors has been changed for the purpose of this chapter by including the masses MIC into the Fourier transform, (3.4):

u(L) =(N M IC)-1/2 L e(K/Ii)Q(Ii) eiq'.r(L), L=(1, K). (10.1) ;.

Also, the normal coordinates Q(Ii) now contain the frequency dependency exp(iwt). Equation (10.1) describes plane lattice waves with complex amplitudes Q(Ii). Quantizing these harmonic vibrations, the lattice Hamiltonian, (9.46) becomes

H =h L w;. [at (},) a (},)+!]. (10.2) ;.

The energy quanta h w;. are the phonon energies whereas at (Ii) and a(li) are phonon creation and destruction operators respectively. Corresponding to the classical normal coordinates Q(Ii), phonon field operators

(2 )1/2 A(Ii)= ~;. Q(Ii)=a(Ii)+at(~) (10.3)

are used [l=.(-q,j); cf. (34.15) with a=+l]. All properties ofa crystal which are uniquely defined by the positions of the nuclei and which can be expanded in the nuclear displacements will be expressed by these phonon field operators. Their formal properties are summarized in Sect. 34 b.

The following commutation relation will be used (see (34.14)):

The delta factor

[A,(Ii), A(/i')] = - i 2 w;.L1(q + q') b jj'

=. -i2w;.b;'I'·

L1(q)=N- 1 Leiq'.r(l)

I

(10.4)

(10.5)

equals one if q is zero or a reciprocal lattice vector; in all other cases, L1 (q) vanishes.

A few thermal average values of phonon operators in the harmonic approximation are discussed in Sect. 34 (see also BORN and HUANG, 1954). In defining the normal vibrations, the anharmonic part of the lattice potential (9.9)

Sect. 10 General lattice potential 127

has been neglected. This interaction Hamiltonian (if expressed in the phonon field operators (10.3)), describes processes in which three or more phonons interact simultaneously. The expansion (10.6) refers to the configuration with minimal potential energy. This configuration is defined by the conditions acJJ/au,,(L) = 0 for vanishing stresses.

With (10.3), the displacements of (10.1) can be written

u,,(L) = 2>,,(KIA) A(A) eiqx(l). (10.7) ;.

Here, for compactness, the new vectors

h1/ 2

8(KIA)=(NMK

2w;.)1/2 e(KIA) (10.8)

are introduced. Substituting (10.7) into (10.6) we obtain

h HA = L, L cJJ(A1···An)A(A1)···A(An)·

n;;;3 n. ;" ... ;'n

(10.9)

The Fourier transformed coefficients of the potential are given by 1

cJJn(A1 ... An) ="h L1 (q 1 + ... + qn) L cJJ ", ... "JL1 ... Ln) B", (K11 A1)··· B"JKnIAn) L, ... L n !ll···an

(10.10)

The delta factor expresses quasi-momentum conservation; it is a consequence of the lattice periodicity which implies that the coefficients in (10.6) depend only on differences of cell indices. The q-dependent coefficients are symmetric in the phonon indices A.

Taking into account the phonon-phonon interaction the lattice Hamiltonian is

H=hLw;.[a (A)a(A)+~]+ L ~ L cJJn(A1· .. An)A(A1) .. ·A(An)· ;. n~3n!;'1 ... ;'n

(10.11)

b) The lattice in a static electric field and under deformation. In order to study in more detail the interaction of radiation with real crystals two features of practical importance should be included in the Hamiltonian: the interaction of a static homogeneous electric field with the crystal and the effect of thermal expansion and external strain.

The interaction of the crystal with a static electric field EO is given by

" 0 V" 00 HEO = - L.,M"E" -2 L.,P,.pE"Ep. " "p

(10.12)

Here M" denotes the dipole moment and P,.p the polarizability of the crystal with volume V. Within the adiabatic approximation, both quantities depend on the nuclear displacements only and are expanded in the same way as the lattice potential:

(10.13)


and

(10.14)

Adding H A and H EO to the lattice potential of the harmonic approximation, the more general potential function

w=<Po+t L u(L)clJ(LE)u(E)+HA +HEO (10.15) LL'

is obtained. Here <Po accounts for a constant potential. The potential energy W is rewritten in the form

(10.16)

where the new coefficients are

Wo=<Po- LM~E~- ~ L~~E~Eg, '" ",p

(10.17)

(10.18)

and

(10.19)

In the following, some basic symmetry properties of the coefficients are used. The coefficients are symmetric with respect to the different sets of indices (Ii' "i' IX;) == (Li' IX;). Because the lattice is invariant if displaced by a lattice vector x(I), the coefficients depend only on differences of cell indices

V¥;,! ... ",JL 1 ••• Ln) = V¥;,! ... ",JI1 , "1 ; ... ; In' "n)

= V¥;,! ... ",jI1 -In' "1; ... ; In_1 -In' "n-1; 0, "n)·

From the invariance of the derivative

au",! (L 1 ) ••• au",jLn)

under infinitesimal rigid translations, the relations

I v¥;',,,,! ... ,,,jL, L 1 .. ·Ln)=0 L

(10.20)

(10.21)

follow. The expansions in (10.13), (10.14) and (10.16) refer to the same configuration as that of the intrinsic potential in (10.6). For a free crystal, the corresponding volume would only be observed if the anharmonicity could be switched off. Because of the zero-point vibrations, the anharmonicity leads to a finite expansion even at zero temperature. In order to describe the effects of

Sect. 10 General lattice potential 129

thermal expansion and external strain, homogeneous deformations of the lattice (BORN and HUANG, 1954; LUDWIG, 1967) are considered.

A general homogeneous deformation is given by the displacements

(10.22)

where e"p are the components of the deformation tensor e. The position vector of the atom L is denoted by x (L). Using the method of long waves, homogeneous deformations are, locally represented by the displacements of an acoustic wave

ii(L)=n sinq x(L) (10.23)

with a wavelength 2nq-l much longer than Vl/3. This wave may be considered as a mode of a larger piece of the crystal in which the volume V is embedded. Differentiating (10.23)

(10.24)

is obtained. Because of the large wavelength, the cosine is nearly one within the volume V and independent of L:

(10.25)

Equation (10.25) implies symmetry relations such as

(10.26)

which are convenient for treating effects of strain. Adding the deformations in (10.22) to the displacements of Eq. (10.7) results

in u(L) = ii(L) + v(L), (10.27)

with

v(L) = L8(K/A)A(A) exp(iq x (l)). (10.28) ;(

By inserting (10.27) into (10.15) a double power series of displacements u and v is obtained. The series which depends only on v, gives the potential of the undeformed lattice expressed by normal coordinates,

where

w,,(A1 ••. An)=-h1 Ll(ql + ... +qn) L ~, ... "JLl···Ln)· B",(K1/A1)···B"jKn/An) L, ... L n Cli •.. an

(10.29)

(10.30)

This definition of the Fourier transformed coefficients is used for each constituent of W: the potential tP, the dipole moment M and the polarizability P.


In addition to (10.29) one obtains with u(L) from (10.22)

I rt:(L)xp(L)eap+t I I rt:a,(LI.:)xp(L)xp,(I.:)eapea'P' (10.31) L. ap L. ap L'. a' P'

while the contribution of the mixed terms is given by

C£l···C£n+tn (10.32) . Ua, (L1)'" uaJLn) Van+ 1 (Ln+ 1)'" van+JLn+m)'

In order to evaluate formally (10.31) and (10.32), the range of forces is assumed to be small compared with the volume extension V 1/ 3 . The important case of Coulomb forces needs a more careful discussion (BORN and HUANG, 1954; COCHRAN and COWLEY, 1967). In this case, Ewald's Theta-function transformation is useful.

Within the approximation of short-range forces, the first term of (10.31) can be reduced to

I rt:(0, K) Xp(K) eap · (10.33) K,ap

This term gives rise to a strain-dependent electric moment depending on the static field EO. The second term of (10.31) can be evaluated as

t· I I Paa,(LI.:)xp(L)xp,(I.:)eapea'p' (1'=0). (10.34) L.ap L',a',p'

Using (10.26), the elastic deformation energy may be written in a more symmetric form given by BORN and HUANG (1954),

t I I [ctc( f3 f3'] e,p ea, P' (10.35) ap a' P'

with

[ctct', f3 f3'] = -t I Paa, (LI.:) [xp(L)-xp(I.:)] [xp,(L)-xp'(I.:)]. (10.36) L,L'

From (10.35) the elastic constants can be derived. To lowest order in eap , the following terms of the mixed series of (10.32) are found:

n=l, m=l: h I ~aP)( -, ,1,°)eap A(,1,°), (,1,O=O,j) (10.37)

ap,j

where

(1O.37a)

for generell nand m:

(10.38)

Sect. 10

where

General lattice potential

Ii It(allh) ... (an/ln)( - ... -, }'l ... }'m) = I ~, ... an+JLl ... Ln+ m) L1 ... L n+rn

al···(Zn+m

131

(10.39)

In order to obtain a final expression, the crystal potential is divided into two terms:

W=~ I I <Pala,(LILZ)Ual(LI)Ua.(Lz)+ V. LILi alai

The first part of W equals Ii -- IW;.A(A)A(A), 4 ),

which, together with the kinetic energy of the nuclei

(10.40)

(10.41)

(10.42)

gives the harmonic Hamiltonian (10.2). V, the second part of W, represents all interactions. For N elementary cells, the general lattice Hamiltonian is then

N

H=Ho+ V=Ho+ I Vn ,

n~O

where the different terms of the general lattice potential are:

Vo=Wo+ I ~(AO)x/l(K)ea/l+~ I [aa',j1j1']ea/lea'/l'+'" K,a/l a/l,a' /l'

and

Here, the q-dependent coefficients in Vn , to lowest order in ea/l' are

Vl (Ao) = Wl (AO) + I Wz(a/l) ( -, A 0) ea/l all

VZ(A 1 Xz) = W;(AI Xz) + I W3(a/l) ( - Al Az) ea/l + ... , all

with

and

(10.43)

(10.44)

(10.45)

(10.46)

(10.47)

They have to be calculated according to (10.30), (10.38), (10.40), and (10.42) with the definitions (10.17) to (10.19).


Working with the above Hamiltonian it should be noticed that the volume of the crystal has changed during the deformation:

V(e) =det 11 + e\V(O). (10.48)

If the crystal is free, the deformation parameters are determined by minimizing the free energy as a function of e~p.

11. Lattice dipole moment.

a) The undeformed lattice. Within the adiabatic approximation the crystal dipole moment can be given as a series in nuclear displacements as in (10.13):

M=Mo+ IM1P(L)u p(L)+t I IM2py(LE)up(L)uy(E)+ ... (11.1) LP LL' py

MO denotes a constant dipole moment of the elementary cell. The linear dipole moment is the term of first order in the displacements; the coefficients M 1P(L) denote the formal charge tensor of the lattice particles. The higher-order dipole moments are physically explained by virtual electronic excitations during the displacements of an atom from its equilibrium position. These moments are connected with deformations of the electronic charge distribution. In the rigidion model, only the linear dipole moment is present. In models which take into account the polarizability of the ions like the simple shell model (Sect. 3), nonlinear dipole moments arise by means of anharmonic forces. For alkali halides these may be derived from a Born-Mayer type two-body potential, while for the diamond structure, because of the equality of the two atoms in a cell, three-atom interactions must be considered (SZIGETI, 1963; WEHNER, 1966). The relation of the second-order dipole moment to the anharmonicity in infrared absorption has been investigated, for example, by SZIGETI (1963), COWLEY (1963), BORIK (1970) and KNOHL (1972).

Substituting (10.7) into (11.1), the dipole moment for N elementary cells (,10 = (O,j)) is written:

M =M8+Ii[IM1 (AO)A(AO)+t I M 2 (Al')A(A)A(1') + ... J. (11.2) j U'

Here the q-dependent coefficients are defined by an equation analogous to (10.16).

b) The lattice in a static electric field and under deformation. Considering, as in Sect. lOb, the lattice in a static homogeneous electric field, an extended dipole moment operator will be used. Adding to (11.1) the dipole moment induced by the static field EO due to the electronic polarizability P,

gives V(e)=detll+el V(O). (11.3)

Here, M n(L 1 •.• Ln) and P,,(L1 ••• Ln) are tensors with the rank n+ 1 and n+2, i.e. they depend on n + 1 and n + 2 Cartesian indices, respectively.

Sect. 12 Lattice and electronic susceptibility 133

Introducing (10.27) into (11.4) a double power series of displacements is obtained which is reordered in the same way as for the Hamiltonian (10.46) with

(11.5)

where

M~ =Mo + Vpo EO, (11.6)

M~ =M~ (AO)=M 1 (AO) + VPl (AO)EO + M z( -, AO) e+ ... , (11.7)

M~ =M~(A):z)=Mz(Al Xz)+ VPz(A1XZ)EO +M3( -, A1XZ)e+... (11.8)

M~ =M~(Al ... An)=Mn(Al ... An)+ VP"(Al ... An)EO

+Mn+l(-,Al ... An)e+.... (11.9)

To be consistent with the Hamiltonian (10.46), the coefficients M' are determined by comparing (11.5) with

M=_aH aEo .

By this way, the coefficients are found to be

and

(11.10)

(11.11)

(11.12)

With (11.11) and (11.12), the thermal average of the crystal dipole moment (11.5) is given as

<M) = - haa~o - L ~ L [aa ° V(Al'" Jon)] <A(Al)'" A(An). (11.13) E n~ln'A E

12. Lattice and electronic susceptibility.

a ) Formal expansion of the susceptibility. The representations of the lattice dipole moment in (11.2) or (11.5) allow the expansion of the susceptibility in terms of normal coordinates A(A).

The lattice part of the susceptibility, (8.24), is decomposed into a double series of retarded phonon Green functions (Sect. 34c) by

1 Gt(MaIMp)= L -,-,

n,m~l n. m. (12.1)

L Ma()'l'" An)M/l(A~ ... A~)Gt()'l'" )'nl),~ ... ),~), A1 ... An. A~ ... ;l.~

where (refer to (34.37))

Gt(A 1 ... )'nIA~ ... ),~)=i B(t) < [A()'l , t) ... A(An' t), A(A~, 0) ... A(),~, 0)]). (12.2)


With the Fourier transforms of (12.1) and (12.2) the lattice susceptibility tensor

h ZL(W)=- L Z(n,m)(w)

V n,m;;;l (12.3)

is obtained where

The phonon Green functions (12.2) are completely determined by the lattice Hamiltonian. They describe the phonons and the phonon-phonon interactions; the photon-phonon interaction is represented by the coefficients of the dipole moment, M(A'l ... An). The electronic part of the susceptibility is given by, (AO = (O,j)),

bX"p =bX~p+ h[2)~p(AO) <A(AO) +! L ~fJ(}'l X2) <A(A1) A(X2) + ... J, (12.4) j Lhh

where the temperature-dependent part is represented by a series of thermal average values.

b) The harmonic approximation. In order to obtain an explicit expression of (12.3) under simple conditions, the anharmonic part H A of the crystal Hamiltonian (10.6) is omitted in this section. In this case, the phonon Green functions and the susceptibilities can be calculated exactly (see Sects. 34d and 38 a):

(X) Linear dipole moments (n = m = 1 in (12.3)). The most convenient way of calculating the retarded harmonic one-phonon Green function, G~)(AIA.'), that occurs in X(1, ll(W) starts from the equation of motion

(::2 +w~) G~0)(AIA.')=2w"bXA'b(t). (12.5)

Using the boundary condition Gt=O if t<O, the Fourier transform becomes

G(0)(111')_ l' 2w" ~ w I\. I\. - 1m 2 ( ')2 UX,," ._0+ W" - W+le

(12.6)

With (12.5), the susceptibility of polar crystals with undamped dispersion oscillators is given by

X~l,l)=LM,,(AO)M (AO) 2w"o. . (12.7) fJ j fJ w~o-(W+le)2

The one-phonon propagators for the phonons A in (12.6) may be written (see (8.43)),

(12.8)

According to (12.7), the absorption spectrum consists of a series of sharp lines corresponding to the number of dispersion oscillators AO with frequencies w"o. In the simple case of an ionic cubic diatomic lattice (where the transverse


optical frequency WTO is the "Reststrahlen"-oscillator) the dipole moments are ex-pressed by (Z e)2

Ma(R)MfJ(R)= h 2 ball' (12.9) Jl Vo W TO

and the susceptibility by

L( )_(Ze)2 1 (1210) XR W - JlVo wio-(w+iB)2· .

Here, Z e is the ionic charge, Vo is the volume of the elementary cell, and Jl is the reduced ionic mass.

f3) Higher-order dipole moments. Nonlinear dipole moments and anharmonicity are closely related. In order to understand the general structure of the theory of absorption in this section the nonlinear dipole moments are retained while the anharmonicity is neglected. For practical calculations, this is justified in the case of crystals with no linear dipole moments such as germanium where absorption is caused in lowest order by higher-order dipole moments.

The calculation of the many-phonon Green functions (12.1) is presented in Chap. G; explicit expressions are given in Sect. 38a. In the harmonic approximation, all Green functions with odd numbers of phonon field operators vanish.

1) Second-order dipole moments. The second-order dipole moment is the most important one in crystals without linear dipole moments. This is the case for all crystals with one atom per elementary cell (e.g. rare-gas solids) and for those with diamond structure. Theoretical investigations of the absorption due to second-order dipole moments were done, first, by BORN and HUANG (1954), LAX and BURSTEIN (1955), and SZIGETI (1960). Their results can be described by the retarded harmonic two-phonon Green function, (38.19),

G(O) ()=[2(W 1+W2)(1+n1+n2) 2(W 1-W2)(n2-n1)]

2,A,A2 W (WI +w2)2-(w+iB)2 + (WI -w2?-(w+iB)2· (12.11)

With this Green function, the susceptibility is obtained

(2,2)( )=1.'" M (A ')M (1 1) '" 2(W 1±w2)((n2+-t)±(n1+-t)) (12.12) XafJ W 21... a 1,1\,2 13/\'1'/\'21... (+ )2_( +1. 0 )2

A,A2 +,_ W1 _W2 W <.0

2) Third-order dipole moments. Three-phonon bands have been observed in diamond, germanium and silicon. They are partially caused by anharmonic effects which may be treated by extending (12.11) to include the anharmonic decay of the two-phonon states. A second cause is the third-order dipole moment for which the corresponding susceptibility X(3,3) will be given here.

From (38.23), the susceptibility is obtained

[3] _1. '" - - - (0) XafJ (W) - 6 1... Ma(Al' A2, A3)MfJ(Al' A2, A3)G 3 ,A, A2A3(W), A, A2A3

G(O) (W)=[2(W 1 +W2 +w3)((1 +n 1)(1 +n2)(1 +n3)-n l n2n3) 3,A,' A2A3 (WI +W2 +W3)2 -(w+iB)2

3 2(WI +W2 -w3)((1 +n1)(1 +n2) n3 - nl n2(1 + n3))] + (WI +W2-W3)2-(w+iB)2 .

(12.13)

136 Expansion theory of susceptibilities and polarizabilities . Sect. 12

This expression is only one part of X(3,3) describing three-phonon bands; the remaining part of X(3, 3) is

1 '" ° - ° - 2 WAD "4 L, Ma(,1 ,,1 1,,11)(2n 1+1)Mp(,1 ,,1z,,1z)(1+2nz) z ( . )z· (12.14) AOA1A2 WAO- W+18

This contribution has the same frequency dependence as X(1,1) (12.7). It should therefore be separated from X(3,3) and added to (12.7) as a renormalization term for crystals with infrared-active dispersion oscillators.

3) Contributions of mixed order. The expression for X(1,3)(W) and X(3,1)(W) can be calculated in the same way as the other contributions. It is

(1,3)_1. '" ,0 ,0 , 1)( ) 2wAo Xap -ZL.,Ma(1I. )Mp(1I. ,11.,11. 2nA +1 Z ( . )Z' ADA WAO- W+18

(12.15)

and

(12.16)

These expressions can be collected together with (12.14) and (12.7) into one expression with a renormalized temperature dependent linear dipole moment:

(1) _ '" - ° - ° 2wAo Xap(w)-L,Ma(,1)Mp(,1) Z ( ')Z' AD WAO- W+18

(12.17)

where M(,1 0) =M(,1 O)+t IM(,1°, A., X)(2nA + 1). (12.18)

A

4) The results of this section may be summarized as follows: in the harmonic approximation with dipole moments up to third order the total susceptibility is given by the following sum (in tensor notation)

(12.19)

where

1(1)(w)=IM(},O)·M(,1°) Z 2wAo. Z' AD WAo-(W+18)

and

The absorption spectra consist of sharp lines represented by Dirac delta functions. The linear dipole moment generates a few lines corresponding to the number of dispersion oscillators. The non-linear dipole moments, M Z and M 3'

give rise to quasi-continuous absorption bands.


When co = 0, the lattice susceptibility of (12.19) differs in this approximation from the isothermal susceptibility, calculated at constant temperature from the free energy, G(E) = - [3- llnZ (see Sect. 8) with

Z =Trexp( - [3(H 0 - ME)). (12.20)

by the following term:

This expression cannot be derived from the retarded Green function in (12.1) since that function does not depend on the diagonal elements of the dipole moment M (see Sects. 8c and 34e).

c) Anharmonic susceptibility. The harmonic approximation for XL is of practical use only in non-ionic crystals like germanium which do not contain infrared-active dispersion oscillators. Even in III-V compounds, or substances like SiC, the integrated oscillator strength of the absorption via the linear dipole moment is comparable with that of the non-linear dipole moment and dominates in the frequency region near the transverse optic frequency COTO • In that case, the frequency dependence of the absorption is strongly determined by the frequency-dependent damping function which might be understood as the inverse lifetime of the dispersion oscillator with co = COTO . This damping is due to the anharmonic coupling of phonons and, therefore, is obtained from all possible combinations of two and more phonons which contribute to a decay of the TO-mode.

In the light of (12.3) for the lattice susceptibility, this means that a representation of the retarded Green function Gw is sought in an anharmonic approximation adequate for the discussion of infrared spectra. Such an approximation is derived from a solution of the equations of motion for G w. Since these functions are essentially products of phonon field operators A(A), the starting point is the equation of motion (see (36.9)); here V1(J'1) =0,

(12.22)

with

(n) _ '"' _I_ V (A) - L., (n -1 , V,JA, A1 ... An_ 1)A(A1)··· A(An_ 1)· }'1 •.• A»-! ).

Here, effects of static external fields, pressure, etc., are neglected. From (12.22) the following equation of motion of the retarded one-phonon

Green function (Sect. 38) is obtained:

(cot -(co+is)Z) Gw(A1IAz) =2coA! ()A!~2 -COAl L V3(X 1 A3 A4)Gw (A3 A4IAz)· (12.23) A3A4

Here, higher-order terms are neglected. In the harmonic approximation, v(n) == 0, (12.23) is equivalent to (12.5). The exact solution of (12.23) requires the determination of Gw (}" },IIA") which

is coupled to higher-order Green functions in an infinite set of equations of


motion. In order to derive a practical solution, this system of equations has to be cut off at a certain order; in practice, this means that a certain Green function of definite order has to be approximated by lower order functions. This is equivalent to a certain diagrammatic expansion of, say, thermal Green functions. The interrelation of different types of Green functions is discussed in Sect. 34. Here, the equation-of-motion representation is used which is generally nearer to a physical interpretation.

d) The anharmonic dispersion oscillator. The most important case for our discussion is that of a single dispersion oscillator. This case is realized in all diatomic crystals such as the alkali halides and the II-VI and III-V compounds and, furthermore, in triatomic crystals of CaF2 structure, where of the two TO modes one is infrared active and one is Raman active.

In this case, (12.23) becomes (the single" Reststrahlen" oscillator is denoted by R) using the notation G",(RIR');:,: GR(w) 6RR,

(w~ - (w + is?) GR (w) = 2 wR - W R I V3 (R, ,1,2) G ",(A X IR). (12.24) A

In order to get an approximately self-consistent solution, as discussed above, the self-energy of the dispersion oscillator is defined by a Dyson equation (38.24) :

GR (w) = GkO)( w) [1 -II R (w) GR (w)J,

where GkO) (w) is given by (see (12.6)):

(0) _ 2 w R GR (w) - 2 ( . )2'

wR - W + 1£

(12.25)

(12.26)

From a systematic discussion of the relations of GR(w) to higher-order Green functions, a representation of the self-energy IIR(w) may be developed in terms of anharmonic parts of the lattice potential up to infinite order. In practice, this discussion is mainly restricted to cubic and quartic anharmonicity.

The equation for IIR(w) reads then:

IIR(w) = I V3(R, A, R) G~~ oP·1 X) I V3(A, A3, A4)<A(A3)A(A4) A A3A4

-t I I V3(R,A3,A4)G",(A3A4IAsA6)V3(As,A6,R) (12.27) A3}q A5),6

+(quartic terms).

The structure of this equation shows that an approximate expression for the irreducible part of the two-phonon anharmonic Green function G w(A, A' I A", A"') is needed in order to obtain an explicit description of the self-energy and hence of the one-phonon Green function GR(w). Different possibilities for such an approximation are discussed in Sect. 38. In every case, they correspond to a certain type of factorization which neglects phase relations of higher order; for example, the random phase approximation may be used.

Before discussing the results of an approximation of this type, the general structure of the formal solution of (12.25) will be considered. With (12.26) the Green function is


(12.28)

where .Q~(W) =W~ + 2WR IIR(w), (12.29)

The complex self-energy IIR is split into a real and an imaginary part,

IIR (w) = LlR(W)- i~(w). (12.30)

Usually, that is as long as the temperature is not too high, ~(w) is small compared to wR. Therefore, a quasi-particle can be defined by the pole of GR(w) at the excitation energy Ii WR, where (cf. (38.27))

w~ == Re.Q~(w =wR)

=w~ +2wRLl R(WR)

=W2(W)W~WR' (12.31)

A self-consistent solution of (12.31) needs a determination of LlR(W) from (12.27). A discussion is given in Sects. 36 and 38. Very often, it is sufficient to assume LlR(WR)~LlR(WR)==Ll~; the next approximation leads to

LlR(WR)~Ll~ [1- a~2 LlR(W)W~WR l (12.32)

and so on (see also the discussion in Sects.30e and 30g). Equation (12.31) defines a pseudo-harmonic frequency wR (Sect. 3) for the dispersion oscillator (R). It is a matter of convenience whether WR is supposed to be a "pure" harmonic frequency at T= 0 (which is of advantage if the interatomic potentials are known as e.g. for rare gas crystals) or wether wR rather is defined as the quasi-harmonic frequency wR(T) (LEIBFRIED and LUDWIG, 1961) which includes the effect of thermal expansion of the crystal. The second approach is preferred since the real potentials are not known in crystals such as the alkali halides. However, information about cubic and quartic potential coefficients at a finite temperature can be obtained from certain experiments. Consequently, all parameters of the theory will be assumed to be quasi-harmonic entities. It should be noted that in the renormalized harmonic approximation (RHA), discussed in Sect. 36 one does not consider the effects of line widths.

The imaginary part rR(w) of the self-energy is a frequency-dependent damping function or inverse lifetime of the dispersion oscillator which determines the structure of the linewidth in the infrared absorption due to the dispersion oscillator with W=WTO '

Replacing GkO) in (12.10) by GR(w), the complex lattice susceptibility of a dispersion oscillator is finally obtained for a diatomic cubic crystal (omitting now is).

(12.33)

The real and imaginary parts of the dielectric susceptibility, including the electronic part XO = Xoa from (12.4), then are given by


'( ) . "( ) (l::o-l::co)w(Of (-2() 2'2 ()) (1234) I:: W +11:: W =l::co+[W(W)2_W2J2+4wi~2(W) W W -W +1 WR~ W. .

This result is very similar to a classical formula

cl( ) (I::o-I::oo)wio (2 2'2 ) I:: W =1::00+[ 2 2J2 2 2 WTO-W +1 WYR'

WTO-W +4w YR

where WTO is the experimental frequency and 'YR is a damping constant which is usually fitted to experimental data near WTO (see Sect. lSi).

e) The damping function. The damping function or inverse lifetime of a dispersion oscillator (R) is determined by the different decay modes for the corresponding phonon with W = WTO' In Sect. 38, results are given for the different terms in the self-energy IIR as obtained by the functional method. Here, only cubic and quartic anharmonicity up to second order in the interaction potentials are considered. This seems to be appropriate for the bulk of experimental data, but cases exist where higher-order terms are necessary (refer to Sect. 15).

The results of Sect. 38 for the self-energy of a single dispersion oscillator may be summarized as follows:

IIR(W)=t I [V4(R, R, A1, x1)-I V3(R, R,R')~ V3(R', AI' X1)J (2nA + 1) Al R' W R' I

-t I 1V3(R,A1,A2WG~~~IA2(W) AI.h

-i I 1V4(R,Al,A2>A3WG~~~IA2A3(w), AIA2A3

(12.36)

where the harmonic Green functions are given in (12.11) and (12.13). Only the third and the fourth term in (12.36) contribute to the damping function rR(W) which has the form

~(w)=nsgnw I a I 1V3(R,A1,A2W

· 2(Wl ±w2)6(w2 -(W1 ±w2f)[n 2 +t±(n1 +t)] +i I 1V4(R,A1,A2>A3W(2=t=1)

.",A2A3

· 2(W1 +W2 ±w3) 6(w2 -(WI +W2 ±w3f)

· [(1 +n 1 +n2)(n3 +t±t)±nl n2J},

where the distribution relation

2wi sgnw6(w2-w;)=6(w-wJ-6(w+wJ

(12.37)

(12.37 a)

has been used. The simplest approximation for ~(w) would be to replace the matrix elements 1V312 and 1V4 12 (generally frequency-dependent) by constants or weakly frequency-dependent functions.

The result of such an approximation is

~(w) ~C3 I P2(W)w~wI ±w,[n2 +t±(n1 +t)]

+c4 I P3(W)w~w,+w2±wJ(1 +n 1 +n2)(n3 +t±t)±n1 nJ (12.38) +.-


where the joint densities Pi are

P2(W)ro=rol±ro2= I I (W 1 W 2)-l b(W-W 1 +W2), etc., (12.39) q*O h*h

while the c;'s are adjustable constants or functions. In this approximation the structure of the damping function is completely given by the joint densities (for a discussion of critical points see, for example, COCHRAN and COWLEY (1967), and the discussion in Sect. 3d).

f) The renormalized dipole moment. The results of Sect. 12d are only applicable to ionic crystals where non-linear dipole moments are neglige ably small. In general, anharmonicity and non-linear dipole moments govern different frequency regimes of the spectra. A detailed analysis of the situation in GaAs will be given in Sect. 17 where both types of couplings are equally important.

The presence of anharmonic couplings and non-linear dipole moments at the same time leads to processes of mixed type which can be analysed in terms of renormalized dipole moments. With the approximations of this chapter (V4+n =0, M3 + n =0, n~I), the discussion can be restricted to a renormalization of the linear dipole moment Mi'

With the help of the functional method, the frequencY7dependent renormalized linear dipole moment of a dispersion oscillator is obtained as, see (38.47):

(12.40)

The contribution of the non-linear dipole moments, especially that of M 2 in the third term, leads to shifts and asymmetries of the resonance infrared absorption near WTO' This will be discussed in Sect. 15.

g) The general form of the lattice susceptibility. The results of the foregoing sections are summarized by giving a quite general formula for the susceptibility in crystals which may have several dispersion oscillators R, but with a negligible coupling between them. In this case, the following results are found:

(12.41)

In this formula, the second and third terms are identical with the corresponding terms in (12.19) while the first term contains renormalized linear dipole moments MR instead of M(R) in (12.18) and anharmonic propagators GR,(w) which change the general form of the resonance part of the absorption from delta functions to rather complicated asymmetric profiles.


h) Coupling of dispersion oscillators. In crystals with several dispersion oscillators with a non-negligible coupling (which is the case, e.g. in ferroelectrics) the absorption formula (12.41) has to be modified. This modification has its origin in non-diagonal elements of the self-energy (12.36). For the case of two coupled oscillators, the effective retarded Green function of one of the dispersion oscillators, say (R l ) is found to be (see (38.35))

where the coupling energy is

IIR1R2 (w) = LlR1R2 (w) - i~,R2 (w)

2WR (12.42)

=t [~V4(Rl,R2,Al,Xl)-;~~./3(Rl,R;,R2) ~Ri V3(R;,A\,X\)] (2n\ +1)

-~ I V3(A\,X2,R\)V3(R2,Xl,A2)G~~~,;)w)

-i I V4(A\, ..1. 2, ..1. 3, R 1) V4(R 2, Xl' X2, X3 ) G~~\,A2AJW). (12.43) A1A2 A3

A corresponding equation holds for the second oscillator (R2).

i) Anharmonic coupling parameters. The anharmonic expansion parameters in wavenumber space are given in (10.10); they are similar to Fourier transforms of the expansion parameters in ordinary space (10.30).

The coupling parameters T:,: .. O:n are usually restricted to cubic and quartic order 'as explained above and to nearest- and second nearest-neighbor interactions in ordinary space except for anharmonic Coulomb interactions which need a particular treatment. Examples of those coupling parameters are investigated in detail in Sect. 15 for the case of cubic crystals, where the dominating role of the short-range cubic nearest-neighbor coupling in infrared absorption will be exemplified.

The cubic and quartic part of the anharmonic potential plays an important role also in the third-order and fourth-order elastic constants, in the mode Gruneisen parameters and some further anharmonic properties of the crystal. Since they are only loosely related to the problem of infrared absorption, the reader is referred to the extensive literature in the field which includes the work of LEIBFRIED and LUDWIG (1963), LUDWIG (1967), and BARRON and KLEIN

(1974).

j) The susceptibility under external pressure and in a static field. Further information about the anharmonic interactions involved in infrared absorption can be obtained by measuring the change of the susceptibility under external pressure and in electric fields. The general treatment of these effects has been outlined in Sects. lOb, and 11 b.

There is a characteristic difference between the effect of a static electric field as compared to that of pressure. While both generally shift and split the frequency states and change the existing dipole moments and polarizabilities, the electric

Sect. 13 Lattice polarizability and Raman scattering 143

field may induce new dispersion oscillators in crystals with inversion symmetry and infrared inactive optic modes. The simplest case is the activation of the Raman modes in diamond, silicon and germanium by inducing a first-order dipole moment (11.7). The electric field with symmetry Coo reduces the symmetry of a crystal, i.e. it destroys inversion symmetry and breaks all crystal symmetries higher than axial symmetry.

The application of pressure corresponds to negative stress in the crystal which in the case of hydrostatic pressure is isotropic. In this case the new equilibrium configuration of a parameter-free crystal is given by a similarity transformation, that means that the elastic response is completely determined by the change of the lattice constant Lla=a(T,p)-a(T, 0). The corresponding change of the volume follows from (10.51):

LI V(P)= V([ea/l])- V(O)

= (det 11 + el- 1) V(O). (12.44)

For a discussion of the influence of external fields on infrared properties refer to ANASTASSAKIS (1980).

13. Lattice polarizability and Raman scattering.

a) Formal expansion of the electronic susceptibility. The formulae for the scattering cross section have been given in Sect. 9. It was shown that it is of similar structure to that obtained for the infrared susceptibility if the retarded phonon Green function for the dipole moment is replaced by an analogous one for the susceptibility {)x induced by the lattice vibrations during the propagation of light. The Raman scattering tensor i is a fourth-rank tensor with the same symmetry properties as that of the tensor of the elastic stiffness constants C. In this section, the electronic susceptibilities {)x are connected with the phonon normal coordinates in order to obtain explicit expressions for one- and multi-phonon processes. We expand {)x in a formal way into nuclear displacements as for the expansion of the crystal dipole moment in Sect. 11. Using the Fourier transformation (10.7) for the displacement, the following description is obtained (BORN and HUANG, 1954):

{)Xa/l(Q) =Po,a/l + hO= ~,a/l(AO, Q) A(AO)+! L Pz,a/l(AI', Q) A(A) A (I') + ... J. (13.1) ;.0 ;.,;.'

This equation is the transform of (10.14). The momentum transfer, Q, may be neglected in those cases, where spatial dispersion plays no role. The first term in (13.1) is the practically constant part Xoo of the electronic susceptibility which contributes in only a dispersive way to the light scattering. In this case, the energy transfer, hD, is zero and the elastic part of the scattered light, the Rayleigh scattering, is obtained. Its contribution to the susceptibility is described in (12.4).

The inelastic scattering is due to all higher-order terms in (13.1) which leads readily to one- and multi-phonon processes. The formal similarity of the expansion of {)x in (13.1) with that of the dipole moment M, (11.2), and their analogous roles in the corresponding retarded Green functions for the Raman


scattering and the infrared susceptibility allows for an adaptation of many results obtained for the susceptibility in Sect. 12 to this section. In exactly the same way as in Sect. 12 we find

and Gw(Aj ... An I Xj ... A~) as in (12.3).

These Green functions describe the phonons and their interactions with one another; the expansion coefficients ~p describe the change of polarizability in an external field due to the electron-phonon coupling.

b) Harmonic approximation. From the foregoing discussion it is obvious that a first approximation for the Raman scattering is obtained by using the harmonic approximation for the Green functions Gw(A1 ... An I A~ ... A~) in (13.2). These functions are described in Sect. 12b.

We obtain the following results (with Q~O):

0:) First-order Raman scattering.

. _" * 1 2WR lap,a'p,(Q)- 1... ~p(R)P'/p,(R)-Im Z (Q . )2 (n(Q) + 1)

R n wR - +1£

= I ~~(R) ~'f3'(R) [(n(Q) + 1) b(Q- w R ) + n(Q) b(Q+ WR )]. (13.3) R

The result is a number of discrete lines in frequency distances wR from the Rayleigh lines with Q=wi-ws = ±wR•

fJ) Higher-order Raman scattering. The results are analogous to those obtained for the higher-order dipole moments in Sect. 12. There is only one difference which has to be considered. The Stokes (Q > 0) and anti-Stokes (Q < 0) processes take place in separated frequency regimes (±wi=IQI±ws) while infrared processes formally have Ws = O.

The following results are readily obtained:

with

iaP,a'p,(Q)=t. I ~~(A1XZ)~'P,(Xj)'z) I j±(njnz) (13.4) }"A2 +,-

j+(n j nz)= [1 +n(Q)] ImG~~)AIA2

= [1 + n(Q)] [(1 + n1 + nz) b(Q- w 1 - wz)

+(nz -n1) b(Q-w1 +wz)]

=(1 +n1)(1 +nz) b(Q-W1 -w2 )

+(1+n 1)n2 b(Q-w1+w2 ), (W 1>W2 ) (13.5)

for the Stokes processes, and

j-(n1 nz)=(l +nz) n1 b(Q+W1 -wZ)+n1 nz b(Q+W1 + w z) (13.6)


for the anti-Stokes transitions. Equations (13.4)-(13.6) describe the quasi-continuous second-order Raman spectra.

Third-order Raman spectra can be analysed in the same way as the third- order dipole-moment infrared spectra (cf. (12.19)). So far, a quantitative analysis of the spectra seems difficult even in diatomic cubic crystals. A more interesting effect might be the renormalization of the first -order Raman tensor similar to the renormalization of the linear dipole moment as described in (12.18). This leads to a shift and an asymmetry of the line shape. We shall come back to this point in the analysis of experimental data in Chap. E.

c) Anharmonic treatment. The anharmonic extension of the theory presented in the previous section is a straightforward problem. Regarding the line shape of onephonon Raman scattering (including the interference with two- and threephonon processes) the results of Sect. 12 c can be used. One 0 btains, then, for a Raman-active oscillator R the scattering tensor i by replacing the harmonic quantities peR) and W R

in (13.3) by the corresponding anharmonic ones:

_ - 1 i,p,"'p,(Q) =~p(R)~, p,(R);- 1m GR(Q) [1 + n(Q)] , (13. 7)

whereP(R) is given by (12.40)(with Mn replaced by p")and GR (Q) by(12.28)in a similar way. As a consequence, in crystals without an inversion center, like the III -V compounds where the single oscillator is Raman- as well as infrared-active, the line width can be determined from both types of experimental measurements. The discussion in Sect. 12 on the frequency dependence of the damping, the coupling of dispersion oscillators, etc., holds correspondingly for the Raman scattering and need not be repeated here. For some interesting cases, a discussion is given in CARDONA (1975) and BIERMANN (1974b).

d) Raman scattering in cubic crystals. The symmetry properties of the Ramanscattering tensors i andP are discussed in detail in Birman's article in VoL XXV /2 b of this Encyclopedia. We repeat here the results for cubic crystals to provide a basis for the model theory presented in the following section. Further information is given in the appendix (Tables 40.3 and 40.4) which will be used in Sects. 18 and 19 in connection with the analysis of experimental data.

The analogy between the tensor of the elastic constants and the Raman scattering tensor, which are both fourth-rank tensors of identical symmetry properties, leads immediately to the three independent components of i:

ill == ixxxx = Hi(I;+) + 2i(F1-m i12 == i xxyy =Hi(r;+) - i(r;-m . _. l'(r+) 144=lxyxy =4 11 25 ,

(13.8)

(13.9)

(13.10)

where the indices x and y denote in an obvious way the scattering geometries. F/, F1i, and F2~ are the three irreducible representations which correspond to isotropic (r;+) and quadrupolar charge deformations. As in Sect. 4, we use here the symmetry notation of Loudon (1964). For comparison with other notations, Table 40.4 in the appendix can be used.

146 Expansion theory of susceptibilities and polarizabilities

The relations inverse to Eqs. (13.8) to (13.10) are

i([;+) = 2 (ill +2id=::6AA

i(Tli)=2(i11 -i12)=:: 12AE and (13.14):

i(I;~)= 4i44 =:: 4..1. f'

Sect. 13

(13.11)

(13.12)

where the A's correspond to the notation by KRAUZMAN (1969). It is useful to relate the components of the Raman scattering tensor i to those of the polarizability tensor P, especially for the analysis of Raman scattering by a macroscopic sample consisting of randomly oriented scatterers. The second-rank tensor P possesses two rotational invariants which are independent of the coordinate system; these are the isotropic part of P,

P(T/) =:: pis =i Tr P =i(l~x + ~y + P.,z)'

and the anisotropic part,

y2 =::(pan)2 = 3 [(Pxx - pyy)2 + (~y - P"z)2 + (Pzz - pxx)2 + 6 (P;y + P~ + p;JJ

=3[P2 (I;i)+···J

(cf. WILSON et aI., 1955). From (3.12) we obtain for a cubic crystal

3 ill = 16 n2 «(5c"Y =i(Px; + Py~ + p"~).

(13.13)

(13.14)

(13.15)

The isotropic part of the polarizability is directly connected with the change in the high-frequency dielectric constant coo. Similarly, y2 becomes .

(13.16)

We shall use these relations in the discussion of spectra of cubic crystals. It should be noted that (13.10) and (13.14) are incorrect if the anti-symmetric

part of P is not negligible. In this case, in these equations Pxy has to be replaced by the symmetric combination ~ . (PXy + ~J, etc. The antisymmetric part is then obtained as

(13.17)

This contribution is down by a factor (Q/wY compared with the symmetric part of ixyXY • Therefore, it should become observable in resonance Raman scattering when this factor is of the order of one.

c) Raman coupling parameters. In (10.14) the crystal polarizability was formally described by a power series in the ionic displacements. The expansion coefficients of this series are Fourier transforms of the Raman polarizabilities ~/l( {,A.J) which have been used in the foregoing sections. The relations between both sets read as follows (see (10.8) and (10.30»

1 ~/l(R)=h I ~/ljL)Cy(KIR),

Ly (13.18)


1 ~p(AIA2)=hL1(ql +q2) L I~Il,Yb(LIL2)8y(KIIAI)8b(K2IA2)

L1L2 yb . exp(i[qi' x(lI)+q2' X(l2)]) (13.19)

and so on, for higher order Raman scattering. Due to the screening of the electron-ion interaction one expects that the Raman coupling parameters perform a rapidly converging series with respect to an increasing distance of neighbors so that nearest-neighbor and second-nearest-neighbor interaction are usually sufficient to describe the spectra. Even for cubic crystals the number of independent parameters for every shell of neighbors is rather high and additional approximations have to be used in order to keep the number of adjustable parameters small. The notation and symmetry of coupling parameters for some high-symmetry crystals are given in Table 40.5.

Of particular interest are the coupling parameters at very long wavelengths, the so-called Pockels or photo-elastic (elasto-optic) coefficients. They can be directly determined with the help of Brillouin scattering and provide therefore an additional check on a particular band or local model of Raman scattering. In this respect they play a role analogous to the elastic constants in lattice dynamics.

In spite of the obvious similarity between the elasto-optic and the elastic constants, symmetry properties are generally not exactly the same. For example, the 21 independent elastic constants have to be compared with 36 photoelastic constants (refer, e.g., to WOORSTER, 1973).

In the case of cubic crystals, the situation is simple. As the three elastic constants Cll , C 12 , and C44 define the bulk modulus and the two shear constants of a crystal, the photo elastic constants Pll' P12' and P44' are related to the volume (r;.+) and the two quadrupolar (r;.~ and r2~) types of Brillouin scattering. We shall discuss the interrelation of the photo-elastic constants to Raman scattering in forthcoming Sects. 14 and 18.

f) Effects of static fields and external pressure. It was discussed in Sect. 12i that the main effect of the application of an external static field E and a pressure P is the breaking of the symmetry of the crystal and that this is most important in crystals with high symmetry, in particular cubic crystals with inversion symmetry. For example, in alkali halides the first-order Raman scattering by optical phonons becomes allowed if a static field is applied.

We summarize a few important points for field- or pressure-induced Raman spectra. As in the case of infrared absorption the effect of hydrostatic pressure on a parameter-free crystal is equivalent to the effect of a volume change, (12.45). Unaxial pressure leads to a splitting of degenerate levels which is sometimes useful for the identification of optic modes and for the analysis of the anharmonic part of the lattice potential.

A very important point is the use of electric fields in ionic crystals with inversion centers at the ion lattice sites such as the perovskites. Here, using Raman scattering, one can observe the infrared active modes due to the fieldinduced polarizability, (11.4). This technique provides a very useful tool in the study of structural phase transitions etc. For a detailed review on field- or pressure-induced effects in infrared absorption and Raman scattering refer to ANASTAS SAKIS (1980).

148 Interpretation of experimental spectra Sect. 14

E. Interpretation of experimental spectra

In this chapter, the general theory developed in the foregoing chapter will be applied to the interpretation of spectra of different types of crystals. We begin with a discussion of the model theory of light absorption and scattering due to phonons. Since for the understanding of the physical origin of these phenomena in crystals the crystal structure and symmetry is important, this chapter will be divided into sections corresponding to specific classes of crystals. The sections are again split into subsections for special families of crystals.

A striking example will be the discussion of ionic crystals with special emphasis on the family of alkali halides which, for a long time, have been the model crystals in the investigation of dielectric properties of insulators. Therefore, the analysis of experimental data will be begun with a discussion of the infrared spectra of ionic cubic crystals. After that will follow a discussion of these spectra in other ionic crystals, in semiconductors and in more complex crystals.

The second part of this chapter is devoted to an analogous investigation of the Raman spectra in crystals. While the kinetics of infrared absorption and Raman scattering have some features in common (energy and quasi-momentum conservation in two-phonon processes, etc., see Table 2.3), the dynamics is usually quite different (light absorption via effective ionic charges as compared with scattering via electronic polarization).

In every case we shall prefer to analyze certain selected examples in great detail instead of giving a broad survey on the steadily increasing number of experimental infrared spectra. The concepts and methods developed are thought to provide a guideline to the interpretation of spectra of similar crystals by comparison with well-understood examples.

14. Model theory of infrared absorption and Raman scattering.

a) General features of infrared and Raman processes. In the foregoing sections the formal theory of infrared absorption and Raman scattering has been developed which describes the lattice susceptibility and the Raman cross section in terms of general (phonon-phonon and electron-phonon) coupling parameters. These parameters correspond to definite orders in the expansion of the crystal susceptibility and polarizability into one-, two-, or multi-phonon processes, and their general properties are defined by the cinematical and dynamical structure of the process under consideration (second rank symmetric dielectric tensor or forth rank Raman scattering tensor). Additional restrictions follow from the space or point group of the crystal which, e.g., reduces in the most favorable case the dielectric tensor to a scalar. We are now left with the hard core of the problem i.e. a discussion of the coupling parameters, namely an investigation of the individual values of them in different crystals, an understanding of the differences between ionic and covalent crystals and of the systematic trends in specific crystal families, such as the alkali halides, etc.

Sect. 14 Model theory of infrared absorption and Raman scattering 149

The best approach to the problem would be, of course, a direct quantummechanical calculation of the parameters. Here, very few first steps have been taken towards explicit calculations, mainly for covalent systems. This situation is not surprising since even for phonons in insulators microscopic calculations are still in its infancies. Therefore, it seems to be reasonable to use model theories with parameters chosen as close as possible to the corresponding matrix elements of the microscopic theory.

In Chap. B, it was shown that a treatment of phonons which initially retains an explicit description of the adiabatic electronic degrees of freedom in the equations of motion before eliminating them eventually in the dynamical matrix, leads to a very satisfactory representation of phonons in different types of crystals. A particular advantage is that it drastically reduces the number of parameters which have to be used in describing dispersion curves of crystals, so that very often a unique relation can be found between the model parameters and independent macroscopic quantities. Such a theory obviously has a much greater appeal for a microscopic theory, since then only the macroscopic quantities, the so-called Landau parameters, have to be derived microscopically without changing the general structure of the theory.

The formalism of Chap. B, which was developed in the harmonic approximation, will be extended in this section to non-linear processes which are still adiabatic in this description. This means, that the electronic coordinates can eventually be eliminated; initially they will be kept in the formalism. The general idea is to express long-range effective ion-ion forces by short-range electron-ion forces and so not only to reduce the number of independent parameters but also to lead to a deeper insight into the origin of the ion-ion forces.

A non-linear shell model was first used by R.A. COWLEY for the description of infrared absorption in alkali halides (1964) and in covalent crystals (1965). Satisfactory results were obtained by KNOHL (1972) and by BRUCE (1973) for the infrared absorption in alkali halides (cf. Sect. 15) while the interpretation of Raman scattering in these crystals by BRUCE and COWLEY (1972) seems to be debatable (KRAUZMAN, 1973). The Raman spectra of cubic ionic crystals have been further investigated with a shell model by HABERKORN et al. (1973), BUCHANAN et al. (1974), and MIGONI et al. (1975).

As mentioned above, there exists an important difference between infrared absorption and Raman scattering. Infrared absorption is based essentially on an induced change in the crystal dipole moment which means that the effective charges of the crystal ions are responsible for the strength of the infrared absorption. Anharmonic short- and long-range couplings between the ions are then the main factors in obtaining a more refined picture of the situation; BornMayer and Coulomb inter-ionic potentials are very good first approximations in that case (Sects. 15-17).

Raman scattering, however, shows a rather different behaviour. The change in the crystal polarizability observed in Raman scattering is first of all a change in the electronic polarizability. In a classical picture of an ionic crystal the main effect of changing the polarizability of an ion is to change its volume i.e. its ionic radius; therefore, as a typical example, the breathing' deformability of the


polarizable ions in alkali halides should show a clear relation to Raman scattering if it is to be more than a lucky simulation of the interatomic forces. This seems not to be the case for the alkali halides where the 'breathing' probably simulates the effect of overlap polarization (refer to Sects. 4 and 18), but may be correct for mixed-valence compounds (GUNTHERODT et aI., 1981).

We may ask for general conditions that differential intra-ionic pol arizabilities play an important role in Raman scattering. They are connected with a completely localized change of the polarizability. Such a change requires very flexible electronic polarizabilities of the ions, that means the electronic wave functions of the ions should be able to re-adjust themselves in response to some displacive change in the ions surrounding. Wave functions of these type correspond to open-shell configurations which are typical for partially covalent crystals with hybridized wave functions forming 'bonds'. For example, Raman spectra of Zn-chalkogenides and Ga-pnictides seem to be mainly due to intraionic polarizabilities (KUNC and BILZ, 1976; refer to Sect. 19).

Even more impressive examples of intra-ionic Raman polarizabilities may be found in hydrides and oxides. It is well known that the normal polarizability of 0-- is not a well defined quantity but depends strongly on its ionic radius (TESSMANN et aI., 1963). This leads to drastic changes of the oxygen polarizability in Raman scattering which manifests itself in strong second order longitudinal optic overtone spectra in earth alkaline oxides, oxydic perovskites, etc. (refer to Sect. 18). The spectra of alkali halides, on the other hand, are missing this part of the spectrum and seem to be governed by inter-ionic differential polarizabilities. This does not contradict the fact that their linear polarizabilities, which are important for the phonons, are well defined in terms of individual ions (TESSMANN et aI., 1963). It means that these ions behave like rather rigid closed-shell electron configurations so that the polarizabilities may be difficult to change without a charge transfer, i.e. inter-ionic processes (Sect. 18, cf. this discussion with that given by PANTELIDES, 1975, where the concept of ionic polarizabilities was questioned).

In the case of strongly covalent crystals such as diamond and its homologues the concept of bond charges and polarizabilities seems to be useful. The bondcharge model for lattice vibrations has been discussed in Sect. 5 c. It can be shown that its non-linear extension leads to a simple and successful description of the infra-red absorption by non-linear dipole moments and of the Raman scattering in terms of bond polarizabilities (Go et aI., 1975). Furthermore, it turns out that bond polarizabilities show an interesting relation to the intraionic polarizabilities defined in a shell model (refer to Sect. 19).

The principal difference between infrared absorption and Raman scattering may also be viewed from the microscopic theory of the dielectric function (Sect. 6). While the effective charges responsible for infrared absorption are essentially connected with the off-diagonal part of the dielectric matrix, i.e., with lattice stability, anharmonicity, etc., which are well-defined in the rigid-ion limit (refer to BILZ et aI., 1974), Raman scattering is based on the diagonal part of the dielectric matrix, i.e. the linear and non-linear polarizabilities of the crystal. A treatment of the problem along these lines does, at present, not exist.


We are now going to a more quantitative description of the general features discussed in this introduction. We recall that in infrared absorption the anharmonic potentials may be described in terms of direct ion-ion potentials so that the discussion of the experimental spectra can be based immediately on the general treatment in the foregoing chapter (Sect. 12) without explicit reference to a non-linear extension of the phonon model. The situation is, as explained above, different in Raman scattering where the energy transfer to the crystal is via the polarizable electrons so that an explicit treatment of electronic pol arizabilities and deformabilities seems to be worthwhile. It will be shown how a model description of the electron-phonon interaction can be obtained from the microscopic theory.

b) Microscopic and model treatment of electron-phonon interaction. In Sect. 9c it was shown that the leading term in the electron-phonon interaction in the energy regime of the electrons (i.e. band gap energies ~eV) is the difference between the adiabatic and the static electron~ion interaction (9.25). Here both terms are defined for some displaced configuration of the ions but the matrix elements have to be calculated with 'adiabatic' and 'static' electrons wave functions, respectively.

In order to understand (9.25) in more detail we discuss first the case of a single electron interacting with a specific ion in a transition where one or several phonons are created. Equation (9.25) reads in this case

with 2 1

Uei = -Ze -I --I· r-R

(14.1)

(14.2)

Here R, r, sand Ro, ro, So denote vectors for the ion, the electron, the quantum numbers of the electronic state for a displaced and the equilibrium configuration, respectively.

We may write (14.2) in the following form:

1 U:'~h (r - R) = - Z e2 S Ir' _ RI (pad(r' - R) - pst(r' - Ro)) d 3 r' (14.3)

with pad(r' - Ro)= pst(r' - Ro).

Now we split pad into a rigid and a deformable part where the rigid part is defined by being equal to pst if the ion is displaced from Ro to R:

p~~(rff -R)= p~~(r' +R-Ro -(Ro +R -Ro)) = pst(r' - R). (14.4)

Equation (14.4) means that the rigid part of pad and pst in (14.3) cancel. Only the deformable part of pad contributes to the electron-phonon interaction:

(14.5)


with r-R=ro+s-(Ro+u). (14.6)

Here u denotes the displacement of the ion and s the corresponding adiabatic shift of the electron. In the rigid ion limit the electron follows the ion motion completely i.e. s = u and Ue~~h = O. This reflects the fact that the electron-phonon interaction depends only on the relative displacements w = s - u.

If we now expand (14.4) into powers of the relative displacements w we obtain, with ao=.ro-Ro, using the notation of (14.1),

Ue~~h = -Ze2<s~Rol ~ f: A [(aoVw)m UeJw=O IsoRo> aOm=l m .

2, ao' w R> = -Ze <soRol-2-+ ... lso ao

(14.7)

= -Ze2 Crss'· w+t 'IJ:/ wa wp+ ... ). (14.8) a,p

The matrix elements f:s',fasp' etc., define dipolar, quadrupolar, etc., electron-ion coupling parameters with respect to one-, two-, etc., phonon processes assisted transitions between the equilibrium electronic states with quantum numbers s and s'. The center of gravity of the electronic charge is R, i.e. the position of the displaced ion. This ensures automatically the translational invariance of the treatment and avoids complicated compensation of large terms if the coupling parameters are calculated for the ions in equilibrium positions ..

Equation (14.8) may be extended in an obvious way to the case of many electrons and ions in a periodic crystal. We obtain then the following 'shell model' representation of the adiabatic electron-phonon interaction:

Ue~~h = ~)cPs,/X wa(L)+t L cPs,/X,s'a' wa(L) wa,(I:) + ... ). (14.9) I,Cl I' a.'

The w(L) denote relative displacements between an electron in a specific cell with index (0, 1), and the ion in the (+ 1)-th cell with particle index K. The coupling parameters depend, of course, to some extent on the details of the model, and this may be the more the case the more a clear separation of the electronic charge density into individual cell or ion contributions becomes debatable.

We may, nevertheless, state a few general aspects of the above-given treatment:

(J() the rigid-ion limit leads to vanishing adiabatic electron-phonon coupling. The parameters of this coupling should, therefore, not be confused with the anharmonic phonon-phonon coupling which is well-defined (refer to (7.129)) in the rigid-ion limit;

[3) a model theory should describe the electron-phonon coupling in terms of quantities (vectors and tensors) which correspond to relative displacements between (effective) electron and ion coordinates. This ensures translational invariance;

y) the most interesting terms are those with L=O, i.e. intra-ionic (or intracell) couplings. They describe a strongly localized change of the polarizability;


15) if the local polarizability is well defined (as in alkali halides) and therefore not easily to be changed, inter-ionic terms should become important. For weakly polarizable crystals the relative displacements of ion cores and electrons tightly bound to neighboring ions can be replaced to a good approximation by relative ion-ion displacements. It may then be expected that a description in terms of 'formal' expansions into powers of ionic displacements becomes quite successful (refer to Sects. 18 and 19).

The discussion of this section provides a basis for a model theory of Raman scattering, given in the next section.

c) Shell model treatment of Raman scattering. We will use in the following a representation of the theory given by STRAUCH (1972, unpublished) which takes advantage of the general properties of Green functions. This shortens the description and shows the interrelations between different types of processes like those between Raman scattering and infrared absorption due to non-linear dipole moments.

We begin our discussion with the important case of second order spectra in cubic diatomic crystals such as the alkali halides. The different processes possible in electronic Raman scattering are shown in Table 14.1.

Here, we restrict the analysis to that of deform abilities with dipolar (I;.5) symmetry which seems to be important for earth alkaline oxides and tetrahedral II-VI and III-V compounds. Extension to other deformabilities can readily be done (see the discussion of silver halides in Sect. 15). The formalism will be applied to the case of defect induced Raman scattering in Sect. 27 f. The notation follows generally Sect. 4, (4.25ff.), where the force constants lP have been labeled with subindices c and e or r referring to ionic cores and electronic degrees of freedom with point symmetry r. For the dipolar deformation of the electronic charge density (shell shift) we used the index s.

a

b

c

d

I

""'9"--f\~ (14.10) Ml=y i e-=tPss

I

~ -VYVO--~--~-~--OVVV" (14.16 )

0-= OJi Vf = tPsFs : OJs I

w~ ~--~~4---{yvvv"(14. 17)

v? V~ i

(14.17 )

Table 14.1. Diagrams for Raman scattering (Notation follows Table 2.1, cf. also Fig. 27.9). a Electronic susceptibility Xao ' (14.10); b First-order (one-phonon) Raman scattering PP), (14.16), (cf. diagram in Table 2.3); c Second-order scattering in terms of two cubic polarizabilities, pP), (14.17) 1.

term; d Second-order scattering in terms of quartic polarizabilities, pP), (14.17) 2. term

154 Interpretation of experimental spectra

The electronic susceptibility Xoo is given in the shell model by

Xoo = YcPs-; 1 Y,

Sect. 14

(14.10)

where the shell charges Y play the role of effective charges with respect to the exciting external electromagnetic field. The inverse propagator cPss reads in the shell model, (4.13),

(14.11)

where S ::;:; R is the short range part of the electron-electron interaction and K the matrix of the local electron-ion interaction.

Equation (14.10) may be compared with the static limit of the quantum mechanical formula for the electronic susceptibility, (7.52)

Xoo=x(O)=e 2 LI<0IR.sln)1 2

V n En-EO

which reads in the closure approximation, with Lin) <nl = 1,

e2 1 Xoo =- --<01(R·s)210)

V LlEeff

(14.12)

(14.13)

with the effective energy gap LlEeff (of the order of the ionization energy of the system) (DALGARNO, 1960).

If we expand (14.10) by introducing unit tensors, I=W1-·wl/w2 with transverse relative shell displacements w 1- (ion displacements are zero), we obtain

(14.14)

which shows a strong analogy to the microscopic expression

(14.15)

We may therefore interpret Y as sort of expectation value for the transverse effective electronic charge in a dipole transition while cPss is proportional to the effective gap.

From these equations we can derive some general features of the change of the polarizability, b X, which occurs in Raman scattering process due to phonons. There are two mechanism to consider:

IX) The effective energy gap might change its value in such a process which means in the model theory that the harmonic potential cPss w2 has to be modified by higher-order terms, such as cPsss w 3 , etc.

f3) The ground-state wave function 10) could change and thereby lead to a change of the dipole matrix elements. In the model theory this corresponds to a change bY of the shell charge.

We mention that this discussion is not sufficient for anisotropic systems where and LlEeff may depend on the direction of the polarization wave. Here a change of this anisotropy has to be considered as a third possibility contained in b X.


In terms of our diagrams, Table 14.1, the first process means a change of the electronic polarization self-energy while the second one corresponds to a vertex correction since the effective charge changes. This is, of course, due to the renormalization of our bare electronic charges via the electron-electron and the electron-ion interaction.

Let us first consider the analytical expression for the self-energy process. With the help of the usual rules for the functional derivatives of propagators discussed in Sect. 34ff. we generate the diagram 14.6 for a one-phonon differential polarizability from (14.10):

P(1)- b Xoo _ 'T.m-1( .m ) .m-1 y. r = (5 wr - ~Y'ss -'PsI's 'Pss , (14.16)

where c[>srs denotes the third-order electron-phonon coupling parameter which is of first order in the phonons characterized by the shell eigenvectors Wr with symmetry r. We may finally replace the wr's by the ionic displacements u with the help of the adiabatic condition, (4.11), but the calculations may be also carried out directly in the 'relative' electron-ion W space since this is the original form of the adiabatic electron-ion interaction.

The second order term can be evaluated in an analogous way and gives:

!Pj2)· (wr wr)r = Y(f>s~ 1 [( (f>srs wr ) (f> s~ 1( (f>sr"s wr ,)

(14.17)

Here, a summation convention with respect to the subindices r, r', rtl is always assumed.

The first term of (14.17) may be reformulated in terms of a phonon modulated shell charge Y:

with (14.18)

We can compare this with the analytical expression for a direct change of the shell charge:

(14.19)

While similar in their general structure, b Yr and b y(l) are different in their energy dependence and should be, therefore, distinguishable in favorable cases.

There are also asymmetrical terms due to a change of Y. For the first two orders they read

(14.20) and

(14.21) with

(14.22)

Here, the arguments for a comparison with the self-energy terms in (14.16) and (14.17) hold correspondingly.

We remark that the application of the Green function technique is simpler than the expansion technique used by R.A. COWLEY (1963) which requires an


elaborate sampling of terms of different order in the nonlinear equation of motions. We note that each term is explicitly translational invariant and that no confusion with anharmonic coupling parameters is obtained.

d) Bond charge and bond polarizability in infrared and Raman processes. In Sect. 5c it was shown that an adiabatic bond charge model gives a very good description of the dispersion curves in diamond and its homologues and even in III - V semi-conductors. One might, therefore, look for a non-linear extension of this model for the description of the infrared and Raman processes in covalent crystals.

As a starting point we use the potentials of the bond charge model as shown in Fig. 5.2. As the analysis of the infrared spectra of many ionic crystals suggests (refer to Sect. 15) it seems to be promising to treat the bond charge Zb as the effective transverse charge with dominating short-range couplings to the neighboring ions. The simplest possibility would be a cubic complement of the ion-bond charge potential <Pi-be' Fig. 5.2, which may be renormalized by long range contributions, in analogy to the situation in ionic crystals. This approximation leads to second order phonon absorption and provides an explicit model for the so-called non-linear dipole moment M(Z) (refer to Sect. 17). This approach has been discussed in detail by Go, BILZ, CARDONA, RUSTAGI and WEBER (WEBER et a1., 1974; Go, 1974; Go et a1., 1975a; Go et a1., 1975b). The main result is that the non-linear dipole moment may be represented by

M(Z)= N Z ",,-1 (f,!3) • u+ U-2 b 'YSS "t'l-be· ' • (14.23)

The result can be obtained directly from diagram in Table 14.1 with the Green function method. The hypervectors U + and U - are built of elements U ;:-", = U"

- U,,' and U ;", = U" + U", - 2 ube respectively. Here, the local displacements in a bond are decomposed into those of even parity, U+ and odd parity, U-. The rather satisfactory results are discussed in Sect. 18. In view of the discussion in the foregoing section one might ask whether the Raman scattering in those crystals is governed by terms

(14.24)

for the first order Raman line with Tz~ symmetry and

1.p(2)(UU) = _1.Z ,tfJ- 1 (?i~4) tfJ- 1 Z U- U-Z r r Z b ss "t'1-be ss b (14.25)

with (14.26)

for the second order process. Physically, this means that the change of polarizability in covalent crystals would be mainly caused by the displacement polarizability of the bond charge corresponding to an oscillating electronic charge center of gravity between the ions. A calculation of the spectra shows that 'this is inconsistent with the observed intensity ratios of quadrupolar (T12' TZ5) divided by volume (T1 ) scattering. Experimentally, the volume scattering is found to be strongest while the theory leads to the contrary result (Go, 1974).

Sect. 15 Infrared spectra of ionic crystals 157

The failure of the bond charge model for the Raman scattering is not unexpected. Even in the harmonic approximation the bond charge model gives only 10 % of the observed dielectric constant (WEBER, 1974) which shows that a rigid point charge approximation for the electronic charge density is insufficient. There are, in principle, two possibilities to overcome these difficulties: The first is a consideration of the missing part of the polarizability by introducing electronic shells at the ion lattice sites. This seems to be a promising idea (refer to Sect. 18) but it would imply a somewhat arbitrary division of the electronic charge into bond and shell charges. The second possibility is a dynamical charge transfer of the bond charge in the direction of its neighboring ions. It seems to be difficult to incorporate this charge transfer into the bond charge model which would loose its attractive simplicity.

An intermediate step into the right direction may be the introduction of bond polarizabilities which depend in first approximation only on the bond length Rb of two neighboring ions.

This description was first used in molecule physics (refer to WOLKENSTEIN, 1941) and for the first order Raman line in covalent crystals by MARADUDIN and BURSTEIN (1967) and FL YTZANIS and DUCUING (FL YTZANIS and DuculNG, 1969; FUTZANIS, 1972). Its extension to the case of second order spectra is straightforward (Go, 1974; Go et al., 1975; TuBINO and BIRMAN, 1975). The general form of the polarizability reads

(14.27)

where rJ. 11 and rJ..l aEe the longitudinal and transverse components of the bond polarizability and Rb unit vectors parallel to the bonds. Equation (14.27) may be used as a starting point for rather different crystals. In the particular case of diamond-type crystals the general structure of the expansion of P into ionic and bond charge displacements is given by

p=pO+pl u+ +}P(2)(U+ u+ + U- U-) (14.28)

with the hypervectors U defined in (14.23). The discussion of the application of the theory to experimental data is given in Sect. 18. We shall then also discuss the relation between bond polarizabilities and the band structure of covalent crystals which leads to an understanding of the systematic trends in the ratio of transverse divided by longitudinal bond polarizabilities when going from diamond to silicon and germanium (Go et al., 1975b). It seems that a similar trend analysis of the Raman spectra in II-VI and III-V compounds can be done in terms of intra-ionic polarizabilities (KUNC and BILZ, 1976, refer to Sect. 18).

15. Infrared spectra of ionic crystals a) Qualitative classification of infrared spectra. [General references: BURSTEIN

(1963), MARTIN (1965), MITRA and NUDELMAN (NUDELMAN and MITRA, 1969; MITRA and NUDELMAN, 1970), BALKANSKI (1973).J

In simple ionic crystals, the infrared absorption is governed by the frequency-dependent damping of the one dispersion oscillator with frequency w = W TO == wR • Such crystals must be at least diatomic, since a monatomic crystal


Fig. 15.1. (a) Lattice modes of an alkali halide in one of the main symmetry directions (100) or (111). wR : frequency of the infrared active dispersion oscillator; wL : longitudinal optic mode frequency. Dashed lines: example of combination bands; summation band TO+TA, difference band TO- TA; (b) reflection R. Full line: without damping; fine dashed line (RCl): with classical constant damping; dashed line (ReI): with anharmonic frequency-dependent damping; (c) real part 8' and imaginary part 8" of the dielectric constant. Full line: 8' with and without damping; fine dashed line (8~1) with classical constant damping; dashed line (8~nh) with anharmonic frequency

dependent damping. (After BILZ, 1965)

with only one atom in a cell does not have any optical branches. It is, therefore, not surprising that diatomic cubic crystals are the most frequently investigated crystals in infrared optics; it will be shown that a careful analysis of their spectra is still rather complicated. Qualitatively, at least at low temperatures, the structure of reflection and absorption spectra in those crystals is determined by critical points of two-phonon summation processes. Figure 15.1 shows the interrelations of dispersion curves wiq), two-phonon combination bands W t (q) ±W2(-q), reflections bands R=I(B-l)(B+l)- l l, and the complex dielectric constant B(W) itself, for a diatomic cubic ionic crystal like LiF, in one of the main symmetry directions (100) or (111).


V (crrrl) 1000 500 250 100 50

100 i : Eo-a

y=O !Y=9~i e;.,~5 i I \: i,' \: :, \:

50 R C·,.) i, \ if '---. i, ;, :, :,

0 .. _ ........... }

AL AR AISbt GaSb t GaAs InSb

LiF NaF NaCI KCI KBr KI CsBr 100

50

t 0

10 20 40 100 200 A(~)

Fig. 15.2. Reststrahlen bands of some alkali halides and ionic compounds. Dashed and dotted lines: reflection spectra of a model crystal for two different damping constants. (After GENZEL,

1967)

It was discussed more than 60 years ago that the structure in the reflection bands of these crystals (see Figs. 15.1 and 15.2) might be related to a frequencydependent damping mechanism of the dispersion oscillator (MADELUNG, 1909). Nevertheless, attempts are still being made to describe the structure of infrared absorption bands in alkali halides or other diatomic crystals by assuming two or three infrared active "oscillators" with classical damping constants. Such a procedure always leads to an incorrect description of the absorption in the wings, as mentioned by GENZEL et al. (1958). Furthermore, it gives an unphysical picture of the lifetime of the dispersion oscillator, measured, say, by infrared absorption. At low temperatures, the damping function is in all crystals a strongly increasing function of frequency, and very often its values are still small near W R, but one order of magnitude higher if W approaches wLO ' As will be seen in more detail below, the maximum of the damping function in diatomic ionic crystals is given mainly by the critical points of the TA + TO combination bands which are situated in many alkali halides about a frequency W ~ 1.5 W R .

Usually, this is not very different from WLO and might, sometimes, give the impression of some "activity" of the longitudinal oscillator at W LO ' It explains, at least at low temperatures, the fact that the lifetime of the transverse optic oscillator at wR is usually one order of magnitude higher than that of the longitudinal oscillator, as measured, say, by a Berreman transmission experiment (1963).

As indicated in Fig. 15.1, the maximum of the damping function is expressed by a minimum in the reflectivity on the short-wavelength side. Figure 15.2 shows reflectivities of diatomic crystals, Fig. 15.3 optical constants of NaCI and KBr.


m'r-------------------------------------------------------,

b

---n

---k

1.0

" \

n,k \ \

0.1 \ \

\ 0.01

\ \

\

\ \ 0.00 I '---'-_L--'-_-"---"-_...l...----'_-'-_L---'

a 50 100 150 200 250

WAVE NUMBER (eml)

Sect. 15

Fig. 15.3a, b. Optical constants of cubic diatomic crystals. (a) nand k in NaCI. (GENZEL et aI., 1965) solid line: classical-oscillator fit; (b) nand k in KI. (BERG and BELL, 1971)


The strong increase of the damping is clearly recognizable near a wavelength which is about 2/3 of the wavelength of the Reststrahlen oscillator (determined by the maximal slope on the long-wavelength side). The qualitative similarity of these spectra suggests that there might exist an approximate scaling law such that, by reducing the spectra of different crystals with an appropriate measure, the experimental data would be made to look rather the same. This is shown in Fig. 15.4 where the imaginary part of the dielectric function of several alkali

. IWI UJ' t1J

10' 1 Ii

-,~

- n - n -

I \ t 1 ii

f - i ~~ -- # •. - ~! ~

t· '~ 10" ~-'-Y~1'" i '/v~·- .. ··· ..

- ~~!f 1\ - //~, 1'\ - ~/ ;: ~ - )/'. "'/1' -/' IN"'? "~

10" /~::;.-;'/?~.~~/'. \/ - \..--../ /" /!#" \~.J' - 't-:C; / /.~ •• : • ' - /..- \) ,.-/, '\ // ,/ -, / " - . - /,/ , f /.,1' /' :/~ \ \

/ 0. /"C," II ~ ___ 1\1

10" 'j '\' 't-:~( , '. - .\'. - ' \ - ' \ ..

, 1,1 - '\'\

'~ i " .

10'4 1\ \ '\ \

- \' -j '. -:\ \ - \'

10" \ - ~,\---- I,· -wr--l--~--'--''-rTTTWr'-'---'--.--r-'TTTTT1-----. __ .-.-n''~\"oTW I I I I II I I I I III I ~I I III

I ---"' g ='Yvw

Fig. 15.4. Reduced imaginary part of the dielectric constant 8, 8"/(80 -8",,), as function of the reduced frequency w/wR in ionic crystals (GENZEL, 1967)


halides is represented. The dielectric function is reduced by co-coo which is proportional to the infrared oscillator strength and, consequently, to the width of the reflection bands. The frequency scale is reduced by the Reststrahlen frequency W R in order to bring all maxima into the same position. One sees immediately that the maximum deviation of the c" curve from smooth behavior (as expected in a classical theory with constant damping) on the high-frequency side is near 1.5 wR due to the summation bands. On the low-frequency side the maximum damping appears around 0.5 wR • It will be seen that this is only partially due to two-phonon difference bands (like TA-TO). At room temperature, three-phonon difference processes begin to dominate the absorption (see Sect. 15t). Furthermore, it is interesting to see that at small frequencies the imaginary part of the dielectric constant tends to become proportional to the reduced frequency. This corresponds to the behavior of a classical oscillator, (12.5), with a damping constant y, and it means that the damping function itself approaches a linear function of W (see (12.33)). It is therefore expected that at very long wavelengths a classical picture may be more appropriate. This is obviously not the case on the high frequency side of the resonance where c"(w) decreases proportional to at least, the seventh power of w. This is completely different from classical behavior and is due to the cutoffs of two-, three-, and multi-phonon bands.

The infrared spectra of other diatomic crystals, like the alkali halides with cesium chloride structure (see CsBr in Figs. 15.2 and 15.4), or MgO, are similar to those of NaCI etc. The same holds for crystals with three atoms but only one dispersion oscillator, such as the crystals with CaF 2 structure. There are, however, differences for more complicated crystals with more than one dispersion oscillator. Striking examples are the ferroelectrics, discussed in Sect. 17.

In the following we will give a detailed discussion of the processes important for the analysis of infrared spectra of the ionic crystals. It seems to be necessary first to discuss separately the different effects of short- and long-range parts of the anharmonic potential as well as those of the non-linear dipole moments, although in practice they are all connected among one another. This separation is not only to avoid confusion; it is done since the short-range anharmonic couplings (including the corresponding part of the Coulomb potential) seem to govern the main part of the spectra while all other effects are smaller corrections.

Since we have seen (Sect. 4) that the lattice potential of simple ionic crystals can be satisfactorily represented in a model theory with deformable electronic shells, we expect that the non-linear extension of the shell models would provide a useful basis for the discussion of infrared and Raman spectra and might be used as a test whether or not the concepts introduced in the harmonic approximation,Chap. B, (such as: "action through the shells", deform abilities, etc.) may be used as starting points for a more general model theory of the lattice potential including anharmonic ion-ion interactions and non-linear electron-ion couplings.

This first part of the analysis implies the quasi-harmonic approximation (LEIBFRIED and LUDWIG, 1961) (Sect. 3d); that means that the parameters of the theory are taken at a given temperature. The derivatives of the potential are


evaluated at the equilibrium position for this temperature and are used to calculate the anharmonic self-energy and related quantities as a function of frequency. The renormalization of the quasi-harmonic frequencies due to the selfenergy shifts is considered here explicitly for the dispersion oscillators only, which interact with the external electromagnetic field. The other phonon frequencies as well as macroscopic parameters like eo and eoo are usually regarded as pseudo-harmonic quantities (Sect. 3d) which contain the self-energy effects and can be practically identified with the measured values at a given temperature T and volume V. The approximate thermodynamic potential is therefore the Helmholtz free energy F(T, V) (Sect. 38b) from which all important thermodynamic relations can be derived.

A treatment of this type is based on the simplifying assumptions that the phonons are well defined excitations and that the anharmonicity does not lead to serious anomalies of the line shapes. These conditions allow, within certain perturbation-theoretical approximations, a direct analysis of the experimental spectra with a limited numerical effort and without a complicated theoretical formalism. The validity of the involved approximations have always to be checked by comparison of the calculated quantities with those obtained from experiments. A discussion of the temperature-dependence of the line-widths of dispersion oscillators and similar important quantities may proceed in this way because in centro-symmetric crystals such as the alkali halides are, the effect of the thermal expansion on the quasi-harmonic parameters can be described at lower temperatures, T < e, by taking into account only the change of volume with temperature (Sect. 15e). This change leads in the simplest approximation for the quasiharmonic parameters to a linear temperature-dependence which allows a determination of purely harmonic values at T = 0 by an extrapolation of experimental values to zero temperature (LUDWIG, 1967). The fortune separation of only volume-dependent quasiharmonic parameters from dynamical anharmonic effects in alkali halides is based on a strong cancellation of cubic and quartic parts of the anharmonic lattice potential in the thermal expansion and facilitates the analysis of the experimental data strongly (Sect. 151).

The discussion of experimental spectra in the following is given in terms of the bulk optical cosntants e' and e", nand k, etc. The evaluation of these constants from the measured transmission, reflectance etc., depends on the scattering geometry and the shape of the crystal. This problem has been discussed in detail by E.E. BELL in his article (1967) of this Encyclopedia and will not be repeated here.

b) The infrared spectra of alkali halides: anharmonic effects. The analysis of the spectra of alkali halides begins with a discussion of (12.34) for the complex dielectric constant e(w), where the anharmonic coupling of the dispersion oscillator to other phonons is considered. The small effect of non-linear dipole moments is neglected in this section but will be discussed afterwards.

The contributions of cubic and quartic anharmonicity to the self-energy of dispersion oscillators are shown by diagrams in Table 15.1 which are related to the corresponding expressions for retarded Green functions in Chaps. D and G.


Table 15.1. Diagrammatic survey of anharmonic processes important in infrared absorption. Notation follows Tables 2.1, 2.2 and 2.3

Diagram Temperature dependence of Contribution to IIR(w) occupation numbers

a 0 Q °

1 +2nJ ocT ,1:

V4

b o-vO:o 1 +nJ +n2 ocT ,1~, n n2-n l

c 0 VGV4

0 (2: ni nk)oc T2 ,1: ,n

d ~ ocT2 ,1R ,rR

V3

e ~. ocT2 ,1~, n

V3

f ~ ocT2 ,1~, n V3

~ g ocT2 ,1~ ,n V3 V3

The analytic expressions for the diagrams in Table 15.1 have been found mainly by following the diagrammatic perturbation theory used first by MARADUDIN and FEIN (1962) and by COWLEY (1963) (see also BRUCE, 1973).

For the calculation of the diagrams of Table 15.1 in terms of the often used thermodynamic Green functions the reader is referred to the corresponding appendix in Vol. 25/2a of this Encyclopedia by COCHRAN and COWLEY (1967). We note that perturbative treatments, especially of the cubic term (Table 15.1 b), have been given by MITSKEVICH (1962a, b), and MARADUDIN and WALLIS (1960, 1962, 1963). BILZ, GENZEL and HAPp (BILZ, GENZEL and HAPP, 1960; BILZ and GENZEL, 1962) showed that a modified form of the Wigner-WeiBkopf treatment used by BORN and HUANG (1954) was able to give correct results. Some minor discrepancies between the different forms of the dispersion formula in the earlier papers were due to neglected interference terms between intermediate states (NEUBERGER and HATCHER, 1963). Dispersion formulae have been derived from the equations of motion of retarded Green functions by


VINOGRADOV (1962) and WEHNER (1966) (see also BILZ, 1966). At the present time there is no longer any disagreement about the general structure of the theory although the relative importance of different mechanisms and interactions is still not completely clear. The various diagrams in Table 15.1 are classified according to their temperature-dependence at higher temperatures. The lowest order cubic and quartic terms (diagrams a and b) are explicitly proportional to T and they give contributions of comparable magnitude to the self-energy II (see Sects. 15e and 15f). This classification is most easily derived from van Hove's perturbation treatment of the anharmonic potential in terms of a dimensionless parameter ;\, (VAN HOVE et al., 1961; COWLEY, 1963; BRUCE, 1973). Following (10.46) we may write: <P A =;\, V3 +;\, 2 V4 + .... The diagrams a and b both belong to the terms 0(;\,2) while terms 0(;\,4) exhibit an explicitly quadratic temperature-dependence at higher temperatures.

A different ordering scheme has been used by GUREVICH and !PA TOV A (1964) and WALLIS, !PATOVA and MARADUDIN (WALLIS, !PATOVA and MARADUDIN, 1966; !PATOVA, MARADUDIN and WALLIS, 1967). They discuss the deviation of the mass ratio m+/m_ in alkali halides from unity as a measure for the difficulty to fulfil the conservation law of energy in a two-phonon decay process as compared with a three-phonon decay process. Their (rather approximate) numerical results for LiF and NaCI so far do not support their assumptions.

The analytic form of the dielectric constant of a single anharmonic dispersion oscillator (12.34) is rewritten for the purpose of this section in a slightly different way (WEHNER, 1966; BILZ, 1966):

e'(w)+ie"(w)-e oo . W2(0) U(W)+1V(W)=_2() 2.2 r()' (15.1)

eo-eoo w w -w -1 WR1R w

where w(w) is the pseudo-harmonic frequency

w2(w)=wi + 2WR LlR(W). (15.2)

Since LlR(W) usually gives a small correction to WR only, attention is focussed on the damping function rR(w) of (12.37), which determines the structure of the infrared absorption. The anharmonic approximation of (15.1) is, as will be shown, a satisfactory description in strongly ionic crystals like LiF. Furthermore, attention is first focused on a discussion of two-phonon bands, and the investigation of three-phonon processes is left for a later part of this section.

The first step is the introduction of only one anharmonic coupling parameter of third order between nearest neighbors. As in the Kellermann-model (Sect. 4a) a central short-range two-body potential q> is used, that is q> depends only on the absolute amount of the distance between nearest neighbors:

q>(lu(U) - u(L-)I) == q>(r), (15.3)

where the indices + and - denote the positive and negative ions in a cell I. There is usually also a contribution from the overlapping negative second

nearest neighbor ions. This contribution is highly dependent on the details of the model (simple-shell model, breathing-shell model, etc., see Sect. 4). It is therefore neglected in a first approximation.


The anharmonic part of the Coulomb potential is usually neglected, as well. The justification for this procedure is found in the fact that the anharmonic derivatives of a potential cx:r- 1 increase much slower than those of the shortrange potential. This varies in the nearest-neighbor region approximately as r- 9

with comparable harmonic contributions (COWLEY, 1963). This argument overlooks the possibility of a constructive interaction of long-range parts of the Coulomb potential in the phonon-phonon coupling (KNOHL, 1970; 1972). It is therefore not sufficient to add to the nearest neighbor short-range parameter the corresponding part of the Coulomb potential (BERG and BELL, 1970; E.R. COWLEY, 1971; BRUCE, 1973). The effects of the long-range potential will be discussed in Sect. 15 g.

In the following we give the detailed formula for the damping function in the cubic approximation. This provides a basis for the comparison of experimental data with theoretical calculations and for the discussion of different approximations given later on in this section.

The third derivative V30 (Table 2.3) of the anharmonic potential cp is (LEIBFRIED and LUDWIG, 1961):

where

cfJap/OK, 0 K, I K') == cfJ ap/Lo, LO, L)

= cp[31 XaXpXy + cp[21(xabpy + xpbay + xybap ),

cp[3] = r - 5 (3 cp' - 3 r cp" + r2 cp"'),

cp[21=r- 3(rcp"_cp'), r=lxl. (15.4)

The Fourier transform of this potential, according to (10.10) is, for the cubic coefficient in the damping function (12.37):

where the tensor K3 has the components:

I.e.

K~p/q) = - 2i sin (qar 0) bpy [cp'" bap + cp[21 (1- baP)]

K~aa(q) = - 2 i sin (qarO) cp''',

K~pp(q)= -2isin(qaro)cp[21,

K~py(q)=O, otherwise.

In this cubic approximation the damping function (12.37) reads then:

ri(w)=nwR L: L: 1V3(RA1A2)1 2

. 2(Wl ±w2) [(n2 +i)±(n1 + i)] b(W2-'_(Wl ±W2)2)

= L: L: L: IK~py(q)fa~/qW +,- A1A2 aPy

(15.5b)

Sect. 15 Infrared spectra of ionic crystals

with the short notation for the eigenvector combinations:

where

fa~/q) = (8; - 8i)a [e; (,11) e; (,12) - ep (,11) e: ()L2)]

= (2~ ) 3/2 (~t2 -~~2 ) a (M 1 M 2)-1/2(WR W1 ( 2)-1/2

. [et eZ - e1 ei]/iy'

167

(15.7)

Here, (10.8) was used in order to make all dependencies of ri on masses and frequencies explicit.

c) Critical point analysis. First we may confirm that the simple dynamical model described by (15.5) is in agreement with the infrared selection rules (see Appendix, Table 40.6). For example, two-phonon combinations are not allowed for q at r or X but are possible at L, with the exception of TO + LO and T A + LA, and all overtones are forbidden. These rules for the symmetry points r and X can be in our approximation derived from the phase tensor K(q) since

On the other hand, the bilinear tensor of the eigenvectors in (15.7):

E+ - (qj 1j 2) == 8+(qj 1) 8- (qj 2) - 8- (qj 1) 8+(qj 2) (15.7 a)

vanishes if j 1 = j 2 because of the two terms of E + - cancel each other in this case. This excludes all overtones. For phonon modes at the point L (2roqa = n) one of the sublattices is always at rest. Assuming for the moment m+ <m , the following zeros are established:

e+ (LIT A) = e+ (LI LA) = e- (LITO) = e- (LILO) =0. (15.8)

Two-phonon combinations are forbidden if in each term of (15.7) at least one e is zero. Thus TO + LO and T A + LA are excluded.

As this discussion shows critical point analysis of the two-phonon spectra can be performed on the basis of the present simple dynamical model if the principal behavior of the eigenvectors is known at these critical points (refer to Sect. 4d).

From (15.5b) it turns out that the main contributions to the damping function come from regions in the Brillouin zone where 2qar 0;:::;; n. This determines a cube in q-space which touches the surface of the Brillouin zone just at the L-points. The four combinations at the L-points, TO + T A, TO + LA, LO + T A, LO + LA, are therefore specially suitable candidates for a qualitative critical-point analysis of infrared spectra in crystals with sodium chloride structure.

The result of such an analysis is shown for LiF in Fig. 15.5. Here, an experimental damping function for LiF is shown as evaluated from experimental


YR

0.35

0.30 (TA+TOlL

I 300 0 K l' "·TOl, 0.25 (TA+LOlL j "'""'" d,,,,,;". 0.20

/ W·y~1 .I

70 0 K /

0.15

Fig. 15.5. Damping function of LiF. (After WEHNER, 1966)

data using (15.1) (WEHNER, 1966): _ 2 V(w)

YR(w)=2WR rR (w)=w (0) U2(w)- V 2(w)' (15.9)

Details of the technique of the evaluation of YR are discussed in Sect. 15.1. The reduced-frequency scale w/wR practically eliminates the quasi-harmonic

shifts between the two curves taken at 70 K and 300 K. The lower temperature data at 70 K are close to zero temperature results since the Debye temperature of LiF is an order of magnitude higher. Therefore, the spectra show hardly any difference bands in contrast to the data obtained at 300 K. The four two-phonon contributions taken at the L-point coincide surprisingly well with the four maxima or 'kinks' in the experimental summation bands. The small values of the absorption bands beyond the two-phonon cutoff demonstrate the weakness of three-phonon summation bands at these temperatures. This does not hold in the region of difference bands. Figure 15.5 exhibits a very strong increase of the damping function about w=O.77wR which cannot be explained by two-phonon difference bands, as will be discussed later. Also, it should be remarked that the four difference combinations LO - T A, LO - LA, TO - T A and LA - TO, taken at the L-point, have frequencies w/wR : 1.29, 0.69, 0.35, 0.24, respectively, and do not show any relation to the experimental data at 300 K.

A further point to note is how the frequency-dependent damping function differs from classical damping characterized by a constant yCl. In order to perform an appropriate comparison, it is assumed that in (15.1) w(O)~W(WR)~WR and 2wRrR (w)=YR(w) is replaced by wyCl. Figure 15.5 shows the corresponding linear function as fitted to the experimental data at 300 K. It


can be seen from the figure that such a description holds only in a very small frequency region around W R and might lead to wrong values of 'linewidths', if these are taken from a larger frequency region.

d) Density of states approximation. Since the phase tensor K(q), (15.5b), qualitatively favors regions with larger q values in the Brillouin zone where the density of states is high, it is expected that a description of the damping function using a combined density approximation (see Sect. 12e) should give satisfactory results. In the case discussed here the damping function, (15.6), may be written

ri(w)~ri,D(w)

=const. L L [(n 2(q)+t)±(n1 (q)+t)] 6(w-lw 1 ±(2 1)· (15.10) +,- qhh W1 (q)W 2 (q)

Figure 15.5 shows a combined density, L(qjlj2) 6(w-(w 1 +(2 )), for LiF as calculated by SMART et al. (1963) at T=OK and derived from Hardy's deformation dipole model. The agreement is very good with respect to the positions of maxima, minima and cutoffs.

It is not surprising that the two-phonon combined density exhibits a close relationship to the important two-phonon critical points in Fig. 15.5. This relationship becomes clearer when the density is expressed in a different representation often used in lattice dynamics (MARADUDIN et aI., 1971):

1 P12(W) = N L6[w-(w 1 (q)+W 2 ( -q))]

q

(15.11)

Here, the density PIZ for any two branches jl and jz is expressed by a surface integral in q-space; Vo means the volume of the elementary cell in ordinary space, and the surface element is denoted by d Sq. The critical points of the density P12 appear at frequencies for which ~(q) equals zero or changes sign discontinuously. The behavior of the different types of analytical critical points Pi in the density of states is given in (3.35).

Clearly, to obtain a two-phonon critical point it is sufficient to combine two one-phonon critical points, that is, to have Vw 1 (q)=Vw z(q)=0 separately for each branch. Therefore, most of the structure of a combined density can be found by inspection of the one-phonon densities, especially cut-offs and gaps are easily determined.

In the case of LiF discussed above, the four frequencies at the L-point correspond to saddle points (T A: PI' LO : P2 , TO: P2) or maxima (LA: P3 or Pz) as can be seen by inspection of w(q) (refer to BILZ and KRESS, 1980). If two saddle points with equal indices come together in a two-phonon critical point, one should expect a corresponding shape for that critical point. In this way, the combinations LA+TO and LA+LO resemble the 'kink' behaviour of the original P2 critical points with a strong decrease in density with increasing frequency. If, on the other hand, two saddle points with different indices com-


bine to a two-phonon critical point, a more pronounced effect may be expected. The reason is that the Taylor expansion of the dispersion relation w(q) near a critical saddle point contains, for type P1' one negative and two positive terms while the reverse is true for a saddle point P2 . In consequence, the combination P1 + P2 can have a fairly strong cancellation of terms quadratic in q, possibly leading to a combination branch very flat in q-space and, therefore, showing up in a density maximum. This seems to be the case for the combinations T A + TO and T A + LO at the L-point in LiF (Fig. 15.5).

The result of this discussion may be expressed by analyzing the following representation for the combined density:

(15.12)

Equation (15.12) defines the function J12 which, at least in alkali-halides, turns out to be varying slowly with frequency. J12 may, therefore, be replaced by a constant in a first approximation. Combined densities, calculated in thisLor in a similar way, give surprisingly good results and may be used for an approximate determination of damping functions (BILZ et aI., 1960; BILZ and GENZEL, 1962). It is evident from the form of (15.12) that the combinations of one-phonon critical points show up in the two-phonon density of states exactly in the form discussed in the preceding paragraph.

The determination of damping functions can be improved by correctly including the frequency factors and occupation numbers for ever q-vector as given in (15.10).

)

o 4

2

5 10 15 w(10 13s-1)-

Fig. 15.6. Two phonon densities, (15.10), for LiF at three different temperatures (KNOHL, 1972)


Figure 15.6 shows the results of a calculation of ri,D in LiF by KNOHL (1972) for different temperatures using a deformable-shell model (SCHRODER, 1966). It demonstrates that the position of the maxima of the two-phonon difference bands is much too low as compared with the experimental data at higher temperatures in Fig. 15.5, while the position of experimental summation bands is well reproduced by two-phonon processes.

It may be concluded from this discussion that careful critical point analysis and density-of-states calculations are useful approximations in many cases where only a qualitative or semi-quantitative knowledge of the damping function is required.

e) The effect of short-range cubic anharmonicity. At the beginning of this subsection, the cubic anharmonic potential coefficient V(RA1 X2), (15.5), was given. This describes the coupling of the dispersion oscillator R to two other phonons A1 and A2 with wave-vectors q and -q. It contains, in the nearest neighbor approximation for a central potential, (15.4), three parameters: q/(ro), q/'(ro) and q/"(ro). Two of them are equivalent to harmonic force constants (see (4.5)):

2

cpll(ro)=-2e A. Vo

Therefore, only the parameter cp"l can be used as a disposable one. The assumption of a central potential automatically implies that first order derivatives of the potential with respect to r influence second order 'force contants' (via the equilibrium condition), first and second derivatives contribute to cubic expansion coefficients of the potential, etc.

The first calculations of optical constants using short-range cubic and quartic (see Sect. 15f), anharmonic coupling parameters were done by Mitskevich for NaCI (1961) and LiF (1962) with rather good success. COWLEY (1963) calculated in the cubic approximation the self-energy of NaI and KBr. An improved version of these calculations was given by COWLEY and COWLEY (1965) and recently by HISANO et al. (1972) and BRUCE (1973). Similar results were obtaned by JOHNSON and BELL (1969), BERG and BELL (1971) for KBr, KCI and KI, KNOHL (1970, 1972) for LiF, and E.R. COWLEY (1972) for NaCl. KNOHL calculated the damping function 1'R (w) (15.9) with one adjustable cubic parameter cp"l using a breathing-shell model (Sect. 4d). The corresponding reduced frequency shift JR (w)=Ll R(w)wi 1 was obtained from a Kramers-Kronig transformation (34.83). The results of these calculations are compared in Fig. 15.7 with the values of the empirical damping function as evaluated from experimental data (Fig. 15.5). The agreement is very satisfactory if one considers the neglect of Coulomb anharmonicity and second-order dipole moments in the evaluation of the empirical functions.

KNOHL obtained good results for the reflectivity of LiF at least at low temperatures (Fig. 15.8). With increasing temperature the differences between theory and experiment increases. The origin of these discrepancies should be found mainly in the neglect of fourth-order anharmonicity. Its influence on the spectra will be discussed in the next section.


0.3 r-----,-----,------,------,

70 K

0.2 •••• Theorie

0.1

a

0 0 5 10 15,

w [1013 5 '1] 20

0.10

"3 0.05 .....................

00

0

70 K -0.05

-0.10

......... M •••••••••••••••••••••••••• _ EXP 3)

•••• Theorie

.... - 0.15

b

-0.20 o 5 10 15 w ~013S-~ 20

Fig. 15.7a, b. Comparison of theoretical and empirical anharmonic (a) damping function YR(W) and (b) frequency shift 0R(W) for the dispersion oscillator of LiF. Solid lines: theoretical curves (KNOHL,

1970, 1972). Dotted lines: empirical values (WEHNER, 1966)

The results obtained by HISANO et al. (1972) for KBr and NaCI at room temperature in the same approximation are shown in Figs. 15.9 to 15.12. They indicate that the inclusion of higher order terms should lead to an improved agreement, in particular near the TO mode.

f) The effect of quartic and higher anharmonicity. If one extends the expansion of the short-range two-body potential cp(r), (15.3), up to the fourth-order, one obtains for the relevant coupling parameters (LEIBFRIED and LUDWIG, 1961)

<1>aPyp(OK, OK, lK', lK')=cp!41XaXpXyXp

+ cp!31(XaXp b yp + Xa Xy b pp + XaXp b py

+ Xp Xy bap + Xp Xp bay + Xy Xp bap)

+ cp!21(bap b yp + bay b pp + bap bpy), (15.13)


to .------,--"'""'=::--.------,--,

0.5

................... .'

0

0.5

.....................

0

0.5

..................

0 0 5 10 15

Fig. 15.8. Reflectivity R of LiF at different temperatures. Solid lines: experimental data (FROH' LICH, 1962, 1964; MITRA et aI., 1966). Dotted lines: one-parameter calculations (KNOHL, 1970,

1972)

crn- I

20

10

o

-10

THz 08

0-4

0~~-------/-/-.. _-__ -.. -__ -__ ----------,:!-~.~/--------~~-------1

~/~ """"

-OA

o

----,-------/".- ''', "'"''

'''--'''/

LO 150

Frequency (em-I)

200 250

Fig. 15.9. The calculated shift and width functions for KBr at 3000 K (HrsANo et aI., 1972, Fig. 4)


em-I THzr--------.--------.--------.--------.-------~

40 12

20 06

. '.,r', __ ,,_""" _____ _

o O~--------------------~~ .. ~_~------~~------~

----_ ... -_ ...... -.... -', ...... _-'-

-20 -0-6

o

, ,

\ I

'~'" \ ,I \ \ .... /'/

TO LO

200

Frequency (cm- I )

400

Sect. 15

Fig. 15.10. The calculated shift and width functions for NaCI at 300 K (HrsANo et aI., 1972, Fig,S)

08

04

?: (a) ~ C; 0 ~ .... '" 0::

08

04

120 200 280 Frequency (cm- I)

Fig. 15.11 a, b. Experimental (full curve) and calculated (broken curve) bulk reflectivity for normal incidence (a) KBr, (b) NaCI at 300 K (HrsANo et aI., 1972, Fig, 10)

where qJ[4] = r- 4 qJIV + 6r - 2 qJ[3] + 3 r - 4 qJ[2] (15.14)

with qJ[3] and qJ[2] given in the cubic approximation, as in (15.4) and r= Ixl.


n

o· 140~"---'--:1~00:-'---'-----'-:'160;:-'---'-'Z~ZO O' 0 140

Frequency (cm-') 100

175

160

Fig. 15.12. Experimental (full curve) and calculated (broken curve) real (n) and imaginary (k) parts of refractive index of KBr at 300 K (HrsANo et aI., 1972, Fig. 11)

The parameter cpIV determines in this approximation the contribution of quartic anharmonicity to the damping function, i.e. the second term in (12.37) (see diagram C in Table 15.1).

Similar to the discussion of the cubic potential after (15.4) we obtain the Fourier transform of the quartic potential coefficient in (12.37) with the help of (10.10):

(15.15)

with

qO=qR=O, ql +Q2+q3=Q,

where the phase tensor K4 has the components (see (15.4))

K;i;;b(QJ=2 cos (QiarO) 6yb

. [6ap 6py cpIV + 6ap (1- 6py) cpl31 + (1- 6ap)(1- 6py) cp121] (15.16)

and the function f4 of the eigenvectors ei reads

(15.17)

with, e.g., (12j)=[B;-Bi]a(B;(Qdl)B;(Q2j2)Bi(Q3j3))' etc. As in the cubic case we prefer here to undo the inclusion of the phonon frequencies W A and particle masses M" into the eigenvectors Il(K I A) and to replace these vectors by the original e(K I A)'S, (10.8). The contribution to the damping function is then given


by (see (12.37))

rt)(w)=j-WR L: (2+1) L: {IK4(q;).j4(eR,el,e2,e3W +, - AI A2A3

. (WR WI W2 W3)-1 2(Wl +W2 ±W3) b(w2-(W1 +W2 ±W3)2)

. [(1 + n1 +n2 )(n3 +t±t)± n1 n2 ]}· (15.18)

In the high-temperature limit, where the thermal occupation numbers ni + tare given by kT/hwi , the last factor in (15.18) becomes

(15.19)

We shall now discuss some of the consequences of (15.18). Let us first comment on the symmetry properties, that means the selection

rules following from (15.18). Since now the phase tensor K4 depends on COS(qi'O) it is apparent that here, in contrast to the cubic coupling case, phonons from the X-point may take part in the three-phonon decay process. A discussion of the eigenvectors ei very similar to that given for the cubic case in Sect. 15c establishes easily that certain three-phonon combinations are not allowed in complete agreement with the results obtained with the help of group theory (see Birman's article, Vol. XXV /2 b of this Encyclopedia) Table 40.7. Since, on the other hand, the number of various allowed three-phonon processes is very large one should not expect that a critical point analysis is as useful as in the case of two-phonon processes.

A second point to discuss is the importance of the different processes described by (15.18). The summation processes (corresponding to the + sign in (15.18)) contribute in particular to the damping function in the high-frequency regime above W~2WLO where the contribution of two-phonon processes is zero. As can be seen by inspection of Fig. 15.3 the dielectric constant e"(w) of the alkali halides begins to decrease with a much steeper slope above a frequency 2.5 wR ~ W ~ 3 wR • This is, at lower temperatures, the only frequency region of the infrared spectra where a quartic part of the potential can be clearly identified by its effect on the damping.

At higher temperatures, the contribution of summation bands increases but more important are then the difference processes (cf. the term with the - sign in (15.18)). They may be divided into different classes according to the examples given in Table 15.2.

The first class describes all the processes where the excited dispersion oscillator with frequency W R survives in the decay process (Table 15.2a). In this case the conservation laws of energy and quasi-momentum can be easily fulfilled if the same but otherwise arbitrary phonon A = (qj) is scattered in the decay process. Very similar scattering processes are considered in the discussion of the line widths of local and resonant modes (see Sect. 24ff.). Since the phase space decreases very fast if the scattered phonon starts to deviate from the wand q values of the incoming phonon, the contribution of diagram 15.2a shows a rather sharp peak at W= WR (see Fig. 15.13). The contribution of this anharmonic 'modulation' of the Reststrahlen-oscillator by other lattice vibrations is there-


Table 15.2. Different types of three-phonon decay processes for dispersion oscillators with roR and q=O

a

b

c

0·06

~ 0·06 N ::r: b .s ~ 0·04

0·02

7'Z rotA) rotA)

3 6 Frequency (THz)

\ 9

Fig. 15.13. Contribution to the width from diagram Table 15.3a, rR4 in KBr at 300 K. Solid line: including q-dependence of matrix elements, dashed line: neglecting this dependence. (From BRUCE,

1973)

fore important at higher temperatures for the line-width of dispersion oscillators (see Sect. 151).

The second diagram of Table 15.2 describes scattering processes where phonons near critical points with low frequencies Wi and therefore relatively large occupation numbers ni are involved. For example, we may imagine three equal TA-phonons in the l'-direction in q-space where three equivalent qvectors, ((IX, -IX, 0), (IX, 0, IX) and (0, IX, -IX)), automatically fulfil the conservation


law of momentum. This leads to strong contributions to the damping function near the zone boundary frequency of TA-modes, at about 0.5 to 0.7 WR, and may explain the strong enhancement of the absorption in the alkali halides when going from low to room temperatures (see Fig. 15.5).

The third diagram in Table 15.2 describes decay processes where all three phonons have very small frequencies as compared with W R• These processes should be important for the absorption at very long wavelengths (see Sect. 15 g).

The first calculation of the fourth order contribution at high temperatures to the damping function of NaCI was given by MITSKEVICH (1961). In spite of his very drastic approximations his result shows clearly the relatively strong contribution near W R• Further discussions of the quartic potential were carried out in particular by JEPSEN and WALLIS (1962), WALLIS, IPATOVA and MARADUDIN (WALLIS, IPATOVA and MARADUDIN, 1966; IPATOVA, MARADUDIN and WALLIS, 1967), Moon (1969, 1970) and BRUCE (1973). In Fig. 15.13 the results for KBr of a high temperature (300 K) calculation by BRUCE (1973) is shown and compared with a calculation by !PATOVA et al. (1967) which neglects the qdependence of the matrix elements of V(RA1 A2 A3). It is obvious that the contribution rR4 to the line-width is most important near W R where the contric bution of ri is relatively weak.

The absolute value of the quartic anharmonic parameter as determined from experimental data depends critically on the details of the theory which is used in the analysis of the data. It seems that often the quartic anharmonicity is overestimated by fitting it to the temperature dependence of the line-width of the transverse optic eigen-frequency or to Born-Mayer potentials. This leads to values of the calculated absorption which are much to high in the low-frequency regime W < WR, where the four-phonon difference processes are dominating the absorption at room temperature (Fig. 15.14). We discuss the problem in connection with that of the line-width in Sect. 151.

As mentioned earlier in this chapter, we note here again that the decrease of the absorption constant in the high-frequency regime is mainly due to a corresponding decrease of the damping function. Equation (15.1) reads for w2

~wi S'(W)~Soo

s"(w)~(so -soo) 2w~ rR(W) w- 4 (15.20)

while the experimental values of s"(w) (Fig. 15.3) show a decrease with at least w- 7 and w- 10 in the two- and three-phonon regimes, respectively. Therefore, the damping function ~ must possess a strong frequency dependency itself. This conclusion is consistent with the result of Sect. 15j that the non-linear dipole moment, which leads to a weaker frequency-dependency, is not dominating the absorption in alkali halides.

The very interesting question arising here is whether a general behavior of the higher-order derivatives of the anharmonic potential can be deduced from an analysis of the experimental spectra in the multi-phonon absorption region. This question has stimulated a great interest very recently in connection with high-power laser windows in the near infrared which require highly transparent and extremely pure infrared materials (see, e.g., PITHA, 1973; MITRA and BEN-

Sect. 15

'3 10'

'" '"

10'

Infrared spectra of ionic crystals

-- Theory ------- Experiment

Q2 Q4 Q6 Q8 I~ ~ ~

X

179

Fig. 15.14. Theoretical and experimental results for the imaginary part of 8(W) for LiF at room temperature. The damping function contains ri and rR4 and is determined with the help of a Born

Mayer potential. (IPATOVA et aI., 1967)

Id + +

+

'10 + E +

..g +. 0 +

f- I;

Z ~ W

0 u:::: -I LL 10 + w + 0 U +

zid 0 i= CL

~ 103 (f) CD <[

104

Fig. 15.15. Frequency dependence of absorption coefficient for KCI. Solid circles and solid line: theory. Crosses and empty circles are experimental data from DEUTSCH and RUDKO (1973) and

Am. Inst. Phys. Handbook (1963). (From NAMJOSHI and MITRA, 1973)


DOW, 1975). Experimentally, it is known for a long time that the infrared absorption in alkali halides shows a nearly exponential 'decrease on the high frequency side of the transverse optic frequency up to the regime of 5- and 6-phonon processes (MENTZEL, 1934; HOHLS, 1937; KLIER, 1957; CALIFANO and CZERNY, 1957). Very recently, measurements in different ionic crystals have been carried out by HorGAN and DEUTSCH (1972) and DEUTSCH and RUDKO (1973). Their results for KCI are shown in Fig. 15.15 together with a theoretical fit by NAMJOSHI and MITRA (1973). Other attempts to explain the exponential absorption behavior are by SPARKS and SHAM (1972), MILLS and MARADUDIN (1973), and by BEND OW et al. (1973). The latter paper gives the most general and thourough discussion of the problem. The general conclusion which can be drawn from all these treatments is that the exponential frequency dependence of the multi-phonon absorption is essentially reflecting the exponential character of the short-range ionic potential which leads to a corresponding decrease of the anharmonic coupling parameters v,,(Ai) when going to higher and higher orders n in the multi-phonon spectra. If the temperature is not too high, the summation process dominate the absorption and the phonon occupation numbers are close to zero. Then, the application of the above argument to (15.20) leads, independent of the details of the calculation, to a nearly exponential decrease of B"(W).

We might ask whether the multi-phonon spectra still show the strong frequency dependence of the damping function which was discussed for the twophonon spectra in the foregoing sections. In fact, Fig. 15.15 exhibits a strong oscillating behavior of O((w) superimposed on the general exponential trend. We may expect that an essential part of the structure in the multi-phonon spectra can be described by the iteration of the dominant two-phonon processes. If we, therefore, start from the most important two-phonon critical point due to a combination (TO+TA)L with a frequency wc~210 cm- 1 we should find shoulders of the absorption constant near n· we with n = 1, 2, ... corresponding to critical points in the 2-, 4-, 6-, etc. phonon spectra. In Fig. 15.15 these frequencies at 210 cm - 1, 420 cm -1 and 630 em -1 correspond apparently quite well to the measured structure.

It seems that the description of the damping function of dispersion oscillators in terms of short-range anharmonicity as given in the previous sections is rather satisfactory. In fact this picture while remaining qualitatively correct is modified if long-range parts of the potential, non-linear dipole moments and some other effects are considered which we are going to discuss in the following sections.

g) Coulomb anharmonicity. The possible importance of the Coulomb potential was discussed by several authors when it turned out, that short-range coupling parameters fitted to infrared data were inconsistent with those obtained from a Born-Mayer potential describing lattice compressibility etc. (BERG and BELL, 1970; E.R. COWLEY, 1971, 1972).

It was shown in these papers that a reduction of the value of the cubic parameter q/", (15.4), by adding its Coulombic nearest-neighbor counterpart improved the agreement between the parameters evaluated from different sources remarkably. It was therefore assumed that some "scaling" of the calculated


self-energy by an appropriate factor might be enough for the consideration of Coulomb interactions (E.R. COWLEY, 1971; BRUCE, 1973). That this is not correct has been shown by KNOHL (1970, 1972) who did explicit calculations of the cubic part of the Coulomb potential for LiF. The treatment is very similar to that of the short-range potential in Sect. 15b and 15e with the essential difference that the phase tensor K 3 (q), (15.6), has to be replaced by a tensor C 3 (q) which is the Fourier transform of the anharmonic Coulomb force constant matrix defined by:

cpCOUL=~ I' ZKZK' 2 LL' IrL -rL'1

= cpr + Cp~ + Cp~ + ...

with Cp~=~ I I Capy(L,I.:) wa(LI.:) wp(LI.:) wy(LI.:) and 3. L.L' apy

w(LI.:) = u(L) - u(I.:).

(15.21)

(15.22)

The Fourier transform of the coupling matrix C is obtained as for the other expansion coefficients of the lattice potential using (10.10) and leads to (cf. (15.4) with xa->darO):

(15.23)

where d; = 0,1,2,3, etc., ro means the anion-cation distance, and d = (da, dp, dy), (d = Idl). The evaluation of the slowly convergent sums in (15.23) over the numbers d; is done most conveniently with the help of theta functions (BORN and BRADBURN, 1947). The calculation leads to parameters C:p~ (q) which describe the anharmonic coupling between cations and anions as does the shortrange matrix K~py(q) but with a modified q-dependence originating from higher-order neighbors. In addition, the coefficients C++(q) and C--(q) describe the anharmonic couplings in the same sublattices corresponding to 2-nd, 4-th, etc., nearest-neighbor contributions which have been neglected in the shortrange case. A table of selected values for the C's is given in the Appendix 40c. We note the symmetry relations

(15.24)

with qa=l=O, qp=qy=O. The effect of the different contributions of the long and short-range cubic

potential as calculated by KNOHL (1970, 1972) for LiF at 70 K are shown in Fig. 15.16. The parameter ((Jill is determined in such a way as to give good mean agreement with the experimental reflection curve (Fig. 15.8). This overestimates ((Jill probably by a factor ~2. Nevertheless, the figure shows that V3 and V3COUL

interfere destructively so that the complete cubic damping function F3 is reduced by a factor about 2 as compared with its short-range value. Similarly, E.R. COWLEY (1971) reports a reduction of about 60 % in his discussion of thermal


10° ..... .. (0, V"', V; I . .

3 . .. (e ,0 ,0 I ........

~ ...... (e ,0, V;l

... 10-1 (e ,V"; V;l

70 K

10-2 ..................... , "444

" , , , ,

10-3 0 5 10 15

w [1013 5-1]

Fig. 15.16. The contributions of long-range Coulomb (C) and short-range (V''', V~') cubic anharmonicity to the frequency-dependent line-width for LiF at 70 K (KNOHL, 1972)

properties of NaCl. HISANO et al. (1972) estimate the reduction for NaCI and KBr by about 25 % while BERG and BELL (1970) evaluate about 50 % for KI. The values of parameters used in different calculations depend on the inclusion or neglect of 2-nd n.n. forces.

As can be seen by inspection of Fig. 15.16 the 'scaling' assumption for LiF is only valid in the frequency regime OJ ~2OJR' Similar values may be expected for other alkali halides. Above this frequency, the complete cubic damping function deoreases much slower than the (scaled) short-range function and finally gives values even higher than the unreduced short-range contribution. This demonstrates that in the high-frequency regime all contributions to the damping functions have to be considered carefully. We shall see that this argument will be given some additional support from the discussion of the non-linear dipole moments (Sect. 15i).

One might ask how large the corresponding correction of the quartic shortrange potential by its Coulombic counter-part may be. From the discussion of this section one can expect, that at lower frequencies, in particular in the frequency region near OJR, a certain reduction of the short-range parameter would lead to a reasonable description of the effect. At higher frequencies, again an increasing deviation from the (scaled) short-range quartic potential may happen, thus leading to a contribution of three-phonon processes which could be somewhat stronger than that discussed in the foregoing section. On the other hand, the essentially exponential short range potential should more and more dominate the anharmonicity of the crystal when going to higher order derivatives in the potential.


h) Absorption at very low frequencies. In the case of long wavelengths, the general dispersion formula (15.1) leads to

(15.25)

and c"(W) 2WRrR(W) 2WR

co-cO') ~[aJ2(O)-wZ]Z -> (i)4(0) rR(W), (15.26)

The absorption is then proportional to the damping function itself. Equation (15.25) is equivalent to a loss tangent of tanb=c"/c'~c"(w)/co which is a convenient quantity in the theory of dielectric losses.

We can estimate the different contributions of cubic and quartic anharmonicity from the results obtained in the foregoing sections. With (15.6) and (15.18) we can write

r R3 (W)OC I I IK3.j3IZ(WRWIWZ)-1[2(Wl±Wz)b(wZ-(Wl±WZ)Z)] +, - Q.h,j2

(15.27) and

rR4(W)OC I I IK4 . rl Z

+, - A1A2 A3

. (WR WI Wz W3)-1 [2(Wl Wz ± W3) b(wZ - (WI + Wz ± W3)Z)]

. [(nz +n 1 +1)(n 3 +1±1)±n1 nz]' (15.28)

For the cubic damping, we obtain at low temperatures (nl~nZ~O) a factor W Z

from the Debye-like density and, assuming for convenience WI ~ Wz ~ w/2, a factor W - Z from the frequency factor so that the first factor w 2 is actually cancelled. In addition, the phase matrix K3 gives, with sin qro,...., q ~ W in the long wavelength region, another factor W Z so that altogether a quadratic decrease of rR3 with decreasing frequency for summation bands is expected. At higher temperatures difference bands become important. In the high-temperature limit

ni +l=kT/nwi (15.29)

the term containing the phonon occupation numbers gives

kT (_1_+_1_) = kT WI ±WZ

n Wz -WI n WZw l (15.30)

which is proportional to W -1 for summation bands under the foregoing assumptions and for difference bands too, if, for example, WI and W z are described by two acoustic phonons with a constant ratio to w:

and WI (q) WZ(q)=C 1 Cz qZocwz.

Apparently this condition is not easily to be fulfilled outside of a very small qregion in the Brillouin-zone. In any case we may expect from this discussion that a higher temperatures the damping function rR3 shows only a more or less linear frequency dependence, rR3 = yw, with a damping constant y. This would cor-


respond to the result of a classical theory where rR has to be replaced by a constant times frequency (Fig. 15.5). Experimentally, such a behaviour has been found in several alkali halides approximately in the frequency regime below 0.1 W R for measurements at room temperature (see Fig. 15.3).

For comparison we regard the quartic damping function rR4 which reads at very low temperatures (nl~1l2~n3~0 in (15.28)):

(15.31)

If we discuss a similar approximation for rR4 as that used for rR3 with WI ~W2~W3~CW and with K4~cosqro-41 at low frequencies we would obtain an 'infrared divergence'

(15.32)

which is in clear contradiction to the experimental situation. The result would be worsened if we would consider the effect of occupation numbers which would contribute, in the high-temperature limit (15.29), another factor w- 2 . The difficulty is that for a three-phonon difference process the condition wiocw for three different phonons, with I(i) qi = 0 and Wi = Ci qi, cannot be fulfilled in general. It follows, therefore, that divergent four-phonon processes are not allowed in the absorption at very long wavelengths.

If we relax the condition wiocw and consider difference processes of the type shown in Table 15.3, the explicit frequency factor becomes a constant. The effects of the occupation numbers and eigenvectors have then to be taken carefully into account. The numerical calculations by BRUCE (1972) for KBr (see Fig. 15.13) and by FISCHER (1974) for AgCI (see Sect. 15m) show the approximately linear behavior of the quartic damping function rR4 at low frequencies and room temperature. The frequency dependence of both damping functions, r; and rR4, turns out to be approximately linear and therefore consistent with the experimental behaviour. At very long wavelengths, i.e. in the millimeter region, the experimental absorption coefficient seems to tend to become a constant (OWENS, 1969). This absorption must be due to impurity and surface scattering because, in the static limit, a constant absorption would be inconsistent with the requirements of the theory that the damping function is an odd function of W (Kramers-Kronig relation, (34.83)).

It seems useful to carry out the further analysis by a discussion of the temperature-dependence of the absorption at a fixed frequency in the long wavelength region. Since the cubic term r3 has an explicitly linear and r4 a quadratic temperature-dependence at higher temperatures one might think that the question can easily be decided. This is not quite the case since the temperature-dependence of the quasi-harmonic parameters 8 0 , 8 00 , wR , and that of the phonon frequencies Wi has to be considered, too. The problem will be discussed in more detail in connection with the line width of the dispersion oscillators (Sect. 151).

The absorption at long wavelengths has been given special attention by Genzel and his co-workers [KLIER, 1958 (LiF); GENZEL et aI., 1958 (NaCl, KCl and KBr); SEGER and GENZEL (SEGER and GENZEL, 1962; SEGER, 1965, 1966) (LiF), HAPP and DOTSCH, 1966 (NaCl)].


STOLEN and DRANSFELD (1965) discussed the low-temperature behavior of some alkali halides in the sub-millimeter region while OWENS (1969) looked into their high-temperature absorption. From these papers, the clear conclusion can be drawn that the three-phonon combinations dominate the absorption at low frequencies. As an example we show the results of SEGER (1965, 1966) for LiF in Fig. 15.17. The plotted curves show the absorption constant 1X=4nkA- 1 which in this regime is proportional to the dielectric constant elf and the damping constant. Above T= 1500 C the quadratic increase of IX is apparent. The strong decrease of the curves at lower temperatures includes, in addition to the decrease of the phonon occupation numbers n;, a quasi-harmonic correction of the parameters (SEGER, 1965).

As mentioned above, at very low frequencies and high temperatures deviations from the theory presented here are observed (OWENS, 1969). It seems that the absorption in the nearly static case shows additional contributions from all types of defect scattering etc. In addition, the assumption of non-interacting phonons in the final states is appropriate at higher frequencies and mean temperatures (collision-free regime) but becomes doubtful if both conditions are not fulfilled. The collisions between phonons have then to be taken into account

12

11

10

9

8

7 :.::: ° 'E 6

u

M

S? 5 . ~I'-

4

3

2

o T(OK)_

Fig. 15.17. Temperature dependence of absorption constant for LiF at long wavelengths (SEGER, 1965). Solid lines: experimental data


by diagrams of the type shown in Table 15.1e, f which are important in phonon transport problems. The reader is referred to the reviews by GL YDE and KLEIN

(1971) and by GOTZE and MICHEL (1974).

i) Non-linear dipole moments. The formal contributions of second- and thirdorder dipole moments to the infrared susceptibility and their interplay with the anharmonic forces was described in Chap. D. The results are summarized in (12.41). For the discussion of this section we represent the different processes in Table 15.3 by diagrams. These diagrams can be derived by replacing the propagator of the dispersion oscillator in Table 15.1 by that of an electronic dipole excitation and the anharmonic vertices v" by the corresponding nonlinear electron-phonon couplings v"n_l (Table 2.3).

The theory has the structure of the linear response of two coupled active oscillators one of which has a very high resonance frequency (electronic oscillator) as compared with that of the other (ionic oscillator). The complete system is driven in the frequency regime of the low-frequency ionic oscillator so that the electronic one is always off-resonant and follows the ionic motion adiabatically. The self-energy of the electronic oscillator has a diagonal part corresponding to IIR(w) for the dispersion oscillator; this part contains 8 00 (the harmonic response) and the susceptibilities 1[2) and 1[3), (12..19), due to the 'direct' effects of the second and third order dipole moments. As has been discussed formally in Sect. 12h, there is also a non-diagonal coupling term in the self-energy matrix of the two oscillators, (12.43), which in the frequency regime of the low-frequency

Table 15.3. Diagrams of infrared processes where non-linear dipole moments are involved. Notation follows Tables 2.1, 2.2 and 2.3

a Ml~ Ml c>---~---o

Wz

b E7 M3 M 3

c

Wz

d Ml V~ - "'--~I


oscillator is conveniently described by a renormalized dipole moment MR(w), (12.40). This renormalization leads to a shift and an asymmetry of the resonance peak of the dispersion oscillator at wR .

The non-linear dipole moments may be discussed in the framework of a nonlinear shell model. This has first been done by R. COWLEY (1963). Using the shell model expressions (Sect. 4) for the potential coefficients V';: (Table 2.2) we obtain the following result for the ratio of the effective non-linear dipole moment M z divided by the anharmonic potential V3:

Ml(O) VZ(OO)-1 VZ1(OAI')

V3 (OU')

(15.33)

where iP sec and iP eee define anharmonic extensions of the shell-core and corecore force constant matrices iPse and iP ee respectively (see (4.32ff.)). If we assume for simplicity that only the negative ion is polarizable and that anharmonic forces are acting through the shells, iP sec:=:::; iP eee' as was found for the harmonic forces, iPse:=:::; iP ee' the following simple result is obtained:

(15.34)

where Xoo means the electronic susceptibility due to the polarizability 11_ of the negative ions (see (4.67)). The same result is derived, in a similar approximation, for the ratio M 3/V4 etc. Since the shell charge Y _ e is usually found near to 3 e in alkali halides, the relative importance of the non-linear dipole moment is determined by the electronic susceptibility of the crystal. This susceptibility is directly a measure for the difference between the transverse effective (Born) charge Z~ and the longitudinal effective (Callen) charge zt, (4.60).

To understand the relative importance of the non-linear dipole moment in more detail, we study the general formula (12.41) in the wings of the infrared absorption

:=:::; 4nh L IMz(A 1 I z)- ~R V3 (RA 1 I z;lz V A1A2 WR-(W1 ±WZ)

. 2(Wl ± w z) c5(wZ - (WI ± wz)Z) [(n z +-t)± (nl +-t)] (15.35a)

and, with (15.33)

'" 4nh Mi L / Yo (x wR 1 )/z '" V wi Al A2 3 MR 1-(Wl ±wz?/wi

. 2(Wl ±Wz) c5(WZ-(Wl ±wz)Z[(nz +-t)±(n1 +-!-)]. (15.35b)


Here, we have neglected small renormalizations and higher-order terms. Equation (15.36) was first derived by SZIGETI (1960) and in the form of (15.35) given by WEHNER (1966).

As can be seen by inspection of (15.37), the relative importance is the highest when w=w1 +w2 is large. In alkali halides, with w 1 +w2~3wR' we obtain for the bracket in (15.35b) at this frequency approximately O.I-X wR/MR. This means that the second order dipole moment becomes important if the "effective frequency" MJX is smaller than 10wR • This frequency was introduced by BORIK (1970). He defined reduced 'anharmonicities' by

v,,* == M ~~ A ) v,,(OA 1 A2 ) n-l 1 2

(15.36)

which are frequencies comparable in magnitude with the infrared frequency wR •

In a short-range approximation the Vn*'s may be treated as constants (including a scaling factors with respect to the Coulomb anharmonicity) and equal to an effective frequency:

From the shell model BORIK deduced

W*=I~WR(1+ZS~Z :::~)I =1!WR(~~-I)1 2 Zp 13 00 +2

(15.37)

(15.38a)

(15.38b)

with Zs the Szigeti effective charge and Zp = Z - Zs the polarization charge, (4.16). In ionic crystals, such as the alkali halides are, Zp~Z, which means that the effective excitation energy hw* for a non-linear dipole moment Mn is rather high as compared with the infrared energy hwR . In this case Mn does not play an important role. The situation is different in III-V compounds where Z <Zp, so that w* becomes comparable with wR, and then the non-linear dipole moment is essential for the understanding of the infrared absorption (see Sect. 17).

In ionic crystals with Zp '"" 0.25 Z, Z '" 1, eo + 2/(13 00 + 2) '"" 2 it follows that

(WJW*)2 <0.01.

Since this factor determines the order of magnitude of the relative contribution of the non-linear dipole moment to the absorption in the two-phonon regime, it is now often assumed that this effect is small (E.R. COWLEY, 1972; BRUCE, 1973).

It has been shown by KNOHL (1970, 1972) that the situation may be different if the Coulomb anharmonicity is taken into account. This destroys the proportionality of the matrix elements of M 2 to those of V3 •

In Fig. 15.18, his results are shown for LiF at 70 K. The frequency dependence of the three different matrix elements in (15.36), rvv==I2:MRV312, rMM == 12: M212, and the interference term, rMv==I2:MRM2 V312 are calculated including Coulomb anharmonicity. If the assumption of proportionality would


1.0

r .... I(r .... )mox

rw I Ir •• ) max

r ... I (r .. v ) max

0.5 70 K

o

0.5

o 5 10 15 20

w [1013 d Fig. 15.18. Contributions to the damping function for LiF at 70 K. The maxima of the r's have been normalized to one by multiplicative factors: 5jwR for Tvv, ~1200jwR for TMV and ~6300jwR

for TMM

701 ,--------,.,...---------,

Fig. 15.19. Imaginary part of the dielectric susceptibility for LiF at 300 K and 7,5 K. Experiment: 300 K. x x x FROHLICH (1962, 1963, 1964), 000 FROHLICH (unpublished), "''''''' GENZEL and KUER

(1956), f::,.f::,.f::,. HOHLS (1937). Theory (only Tvv): + + + 7,5 K, 000 300 K


be correct, all three curves should fall into one. Clearly, the differences become important at frequencies w~2.5 W R where three-phonon processes cannot be neglected (Fig. 15.19). It is apparent that one should take three-phonon processes into account when calculating the absorption in highly polarizable materials.

j) The effect of ionic polarizability. In Sect. 4 it was discussed how the dipolar models consider the polarizability of the crystal. In the shell-model a special assumption is that short-range forces act through the shells. The physical background of this assumption is the small overlap between the ions in alkali halides. It makes, therefore, little difference whether the anharmonic potential is defined between the ion cores or between the shells since the eigenvectors of both particles are still very similar and the phonon line widths due to the cubic part of the potential are governed by the density of states and the phase matrix K (Sect. 15e). The main effect is a small renormalization of the matrix element V3 (KNOHL, 1970, 1972; E.R. COWLEY, 1971; BRUCE, 1973).

It is clear that the effect should increase when going to more polarizable crystals and such with strongly overlapping charges (see the discussion in Sect. 16). Equally, on might expect that the Raman cross section is more sensitive to a proper description of the electronic "shell" eigenvectors since this effect is essentially based on the polarizability of the crystal (see Sect. 18ff.).

We mention, that there is, so far, no clear evidence for an anharmonic effect of other deform abilities, e.g. the breathing deformability, in the infrared spectra of crystals (KNOHL, 1970; BRUCE, 1973). An indirect effect of the quadrupolar deformability of silver ions in silver halides has been discussed by FISCHER (1973). The lowering of the TO branch near the L-point brings the maximum of the two-phonon damping function nearly in coincidence with the transverse frequency wR. ('Fermi resonance').

k) Final states interactions of phonons: anharmonic broadening and bound states. In the foregoing sections of this chapter, the phonons in the final states were always described as quasi-particles with infinite lifetime. Aside from the dispersion oscillators, the effect of anharmonicity on all the other lattice vibrations was only considered by introducing renormalized eigen-frequencies. This is, of course an approximation and one may ask for the effect of the finite lifetime of the phonons or more general anharmonic effects on the infrared spectra. Some corresponding diagrams are shown in Table 15.1 and they demonstrate different types of' final states interactions':

Il() Diagram 15.1g: broadening of the spectra by the individual lifetimes of the phonons;

/3) Diagram 15.1 e: dynamical interaction between different two-phonon combinations;

y) Diagram 15.1: frequency-dependent corrections of the V3 matrix elements (vertex corrections);

(5) Diagram 15.1 d: interaction between two- and three-phonon bands. Process Il() seems to be the most important one leading to a smearing out of

sharp features ('erosion' of critical points). There is a striking example in the case of the I-phonon sideband of the impurity-induced U center mode in alkali


halides (ZEYHER and BILZ, 1969) (see Sect. 30 k). Here, the line-width of the local mode leads to a shift of frequencies in the sidebands and a partial filling of the phonon gap in KBr. Similar effects have to be expected in the phonon spectra of perfect crystals but, at the present time, evidence is restricted to the fact that calculated spectra quite often show a sharper and stronger oscillating structure as is exhibited by the measured ones (see the discussion of AgCl in Sect. 15m). In practice, the finite lifetime of the phonons can globally be taken into account by keeping 8 finite in the Green function (see, e.g., (12.11)). This means a representation of the delta-function by a Lorentzian or a histogram and is usually used to create a smooth curve without spurious features. But it might be regarded as a simple approximation to the phonon-lifetime effects and then the absolute value of 8 influences the determination of the shift function of the dispersion oscillator (see Sect. 151). Typical values of 8 are near 0.1 THz, i.e. about 2 % of the frequency of dispersion oscillators (see, e.g., BRUCE, 1973).

Process 13) is related to the problem of bound states which was first recognized in connection with certain sharp features in Raman spectra (see Sect. 19). The problem has been discussed recently by BRUCE (1973) who investigated the possible influence of Diagram 15.1 e on the two-phonon spectra of KBr. The contribution to the complex self-energy of the dispersion oscillator (37.45) is:

LlR(w)-irR(w)= +± I V3(RAIA2)GH1A2(W) Al ... A4

( 15.39)

As has been shown by Bruce, the calculation can be strongly simplified by exploiting the separability of V4 in this particular case. The calculations are easily extended to that of a diagram where the single vertex V4 is replaced by the infinite series of V4 'chains'. The result of the calculation with <pIV determined from the line-width of W R (Sect. 151) is shown in Fig. 15.20. The effect might become visible in the high-frequency regime near wL . BRUCE' (1973) analysis of a thin-film experiment in KBr remains, however, doubtful since he did not consider the influence of Coulomb anharmonicity and second order dipole moments which are influential in the same frequency regime (Sect. 15i).

I) Line widths of dispersion oscillators and temperature-dependence. In the foregoing section, the infrared absorption was discussed at a given temperature and attention was focussed on the frequency dependence of the spectrum. Here we come to the problem of the temperature dependence at a given frequency which was briefly mentioned during the discussion of the absorption at long wavelengths. The most interesting analysis is certainly that of the resonance frequency wR and, to same extent, that of WL which can be carried out by a thinfilm technique (BERREMAN, 1963). The treatment should establish the consistency of the foregoing investigations of the dynamical behavior with the thermodynamic properties of dispersion oscillators.

A comparison of theoretical and experimental values for the line-width and frequency shift of the resonance frequency requires a careful evaluation of the experimental data. Usually, one neglects the frequency dependence of the


0·16

2 4 6 8 a Frequency (THz)

01r-----~~~----~--_7--------~

- -0·03 :!i:! t;

;::: ~ '" -0·09

-0·15

b 2 4 6 8

Frequency (THz)

Sect. 15

Fig. IS.20a, b. Contribution to the self-energy for KBr at 300 K from the Diagram 15.1 e (_. -) and from the corresponding infinite series ( ... ). (a) Width r(w); (b) shift L1(w). (BRUCE, 1973)

damping and assumes a classical formula as given in (12.35). In classical dispersion theory it follows that W8" or w- 1 8" shows a resonance exactly at W =wTO and a fit to the reflectivity with 80 ,800 , YR and wTO again should exhibit rather precisely W TO • In an analogous way WLO may be obtained by determining the resonances of W8,,-1 or W- 1 8,,-1. The classical approach is applicable to many alkali halides at room temperature where rR(W) often turns out to be a rather slowly varying function of W (see Fig. 15.4). At low temperatures rR shows generally a drastic increase in the immediate neighborhood of WTO so that the procedure becomes questionable.

The further analysis is most easily carried out by using the reduced dielectric function of (15.1) (WEHNER, 1966; BILZ, 1966):

8'(W)-800 W(0)2 [w2(W)-W2] U(w)

8 0 - 8 00 [W(W)2 - W2]2 +4yi(w) (15.40)

V(w)= 8"(W) = W(0)2 YR(W) 80 -800 [W(W)2-w 2Y+4Yi(w)

(15.41)


or

(15.42)

(15.43)

with, (15.2),

w2(W) = W~ + 2WR LlR(W)::::: [WR + LlR (W)J2, if LlR ~WR' (15.44)

In order to obtain the resonance frequencies we apply the condition that the real part of the response function should disappear at a resonance, or,

(15.45)

I.e.

(15.46)

In a self-consistent treatment, wR =wTO and LlR(WR)=O but usually WR is chosen to be the unrenormalized harmonic frequency so that LlR(WR) describes explicitly the anharmonic contributions to the resonance frequency.

Equation (15.46) leads to

(15.47)

This is identical with the classical condition if 'YR = W . const. which is in general only approximately fulfilled. The measured and calculated peaks of 8/1 can usually be brought into coincidence by a slight re-adjustment of the harmonic frequency W R .

The dispersive part of the shift function

LlR(W)::::: [w(w) -we CYJ)J + [we CYJ) - wR]

can be determined via a Kramers-Kronig relation:

-2() -Z( )_ 2pooS 'YR(W/) 'd I W W -w 00 - -- , W W

now 2_WZ

2 00 V' w'dw' = -WZ(O);P ~ U '2 + V'2 w'Z_WZ'

( 15.48)

(15.49)

(15.50)

A frequency-independent contribution to the shift function follows from quartic anharmonicity in first order (see diagram 15.1 a and (38.10))

W(OO)-WR =Ll~+ ... =fL V(RRJeI)(2n" + 1)+ .... (15.51)

" In Fig. 15.7 the determination of 'YR(W) and LlR(w) was shown for the case of LiF. Usually, this procedure requires more data than available and, therefore, is often carried out using simple approximations.

The foregoing discussion gives a consistent evaluation of the measured optical data in terms of a frequency-dependent self-energy at given temperature and volume and provides the necessary basis for the investigation into the


temperature and pressure dependence. We note that effects of the non-linear dipole moments were not considered in this paragraph.

In Sect. 10 the general effect of a homogeneous deformation was analyzed. If we start from the lattice at T = 0 K with vectors XO (L), then at a finite temperature T the lattice point L is displaced by, (10.22),

uT(L) = e(T) xO(L) (15.52)

where the tensor e describes the effect of thermal strain. From this we obtain the quasi-harmonic force constant in the lowest approximation (Diagram 2.2e)

(15.53) a"

which gives the quasi-harmonic contribution to the shift:

E h ~ - T LlR ="2 ~ ~p( - R, R)uap + ...

ap (15.54)

and, with (10.42)

~p( -R, R)=~ a~' rJ>aa' a" (LI.; I.;')

. exp {i [XO(I.;') - XO(I.;)J Ba,(K' I R) Ba,,(K" I R) xp(L)}. (15.55)

This may be related to the mode Grtineisen parameter

Y,,= -~ [8InwI] =KT [8 In wI] 3 8ln V T 8p T

(15.56)

2 -~ -3 L ~a(-R,R),

a

(15.57)

where KT is the isothermal compressibility. Since

T hKT ~ 1 ~ ea{J~-3- ~ y"w"(n,, +z)uaP (15.58)

we obtain for

(15.59)

In this approximation Lli has the same temperature dependence but usually opposite sign as compared to LI~, (15.51), so that some compensation effects between the two terms may be expected.

In order to obtain the complete contribution to the shift function we have to add to (15.53) the real part of the self-energy IIR(w) i.e., with (15.46),

(w~f = wi + 2 wR [LIR (w~) + LliJ

(15.60)

Since the resonance frequency w~ possesses an explicite and an implicite temperature dependence a reliable calculation of w~ looks rather complicated.


10,----------------.------------~

Pressure domain ! Temperature

8

6

~ x 4

o

-2

i domain I

• LiF (TO) o NoF (TO) t;. NoF (LO) o ZnS (TO)

195

Fig. 15.21. Pressure dependence and temperature dependence of the infrared resonance frequency of LiF, NaF, and ZnS (MITRA et aI., 1967)

The problem of the temperature dependence of the shift function and the line widths of dispersion oscillators was discussed in particular by R. COWLEY (1963), !PATOVA et al. (1967), LOWNDES (1970, 1972), E.R. COWLEY (1971), BRUCE (1973), and FISCHER (1974). The analysis is facilitated by the fact that the experimentally measured frequencies are not very different from the quasiharmonic frequencies. One can therefore separate out the volume dependence from the general change of shift and line-width. Using the interrelation between the volume and the pressure dependence of WR , (15.56), one can analyse the problem by inspection of a double logarithmic plot of wR (V) as shown in Fig. 15.21 for LiF (MITRA et al., 1967). Here the effect of pressure on the volume-reduced relative shift of wR is shown on the left-hand side of the figure while the temperature effect is shown on the right-hand side. The deviation of the curve in the temperature domain from the extrapolated pressure domain line shows the shift due to the anharmonic self-energy of wR • The effect is not large in LiF but seems to be very small in KBr (POSTMUS et al., 1968). This is indicative for a strong cancellation of the different anharmonic contributions to LlR(WR). It has been observed in NaCI (E.R. COWLEY, 1972), KBr and LiF (Moon, 1969, 1970) and can only happen if the different terms in LlR(WR) have approximately the same temperature dependence. This holds for the two lowest order diagrams in Table 15.1, i.e. the quartic anharmonicity in first order and the cubic anharmonicity in second order which both show a linear temperature dependence at higher temperatures. As a consequence, the contribution of the quartic anharmonicity in second order (Diagram 15.1 c) should be small since it leads to a temperature dependence ex T2.


This result seems, at the first glance, to contradict the experimental observations on the temperature dependence of the line-width rR (wR), which exhibits a nearly T2 behavior at room temperature in several alkali halides (LiF: HEILMANN, 1958; NaCl: HAss, 1960, etc.). Attempts have been made to consider the quartic term in the damping function for the explanation of the temperature dependence in particular by JEPSEN and WALLIS (1962) and !PATOVA et al. (1967).

It was suggested by Moon (1969) and E.R. COWLEY (1972) that the apparent temperature dependence close to T2 was, in fact, essentially due to the implicit temperature dependence via the quasi-harmonic frequencies and force constants of the phonons in the decay channels. This argument would be consistent with the above given analysis of the shift function. The situation can be discussed most conveniently by analysing ri3)(wR ) with the help of (15.6) and (15.7). The question is how to describe the temperature dependence of the phonons with frequencies WI and W 2 • Two simple alternative assumptions are

1) to assume wi(T)ex::wR(T) (Moon, 1969), which perhaps overestimates the effect with respect to the acoustic phonons (BRUCE, 1973) since these exhibit a much smaller temperature dependence. The result in this case is (Moon, 1969) for the fractional width

(15.61)

which seems to be very consistent with his measurements of KBr. E.R. COWLEY (1972) could reproduce the temperature dependence of rR/WR in NaCI as T1.6 fairly well with a similar analysis;

2) to include only the temperature dependence of wR and that of the cubic anharmonic coefficient. This leads to (BRUCE, 1973)

rR/WR ex:: wR (T? . T, (15.62)

which describes a temperature dependence less than linear so that quartic processes are inevitable.

We think that the treatment by Moon and COWLEY is close to the real situation for several reasons. Firstly, the temperature dependence of the w;'s even if smaller than that of wR should not be neglected. Secondly, the quartic processes near WR are mainly difference processes (see the discussion in Sect. 15t) so that their temperature dependence is even higher than T2 at lower temperatures. Thirdly, a self-consistent treatment of phonon frequencies leads naturally to temperature-dependent renormalizations of TR , WEHNER (1966, see (38.59)ff). A renorma1ization of the final states in the quasi-harmonic approximation leads to further factors W;{T)-1 in front of the line-widths. But the most important point might be the broadening of the final state phonons which leads to a replacement of the b-functions by the imaginary part of the proper two-phonon Green functions. The problem was touched upon by R.A. COWLEY (1971) and BRUCE (1972) and recently discussed in some detail by FISCHER (1974). He formulated a consistent extension of the pseudo-harmonic approximation where the renormalized frequencies Wi of the phonons are supposed to contain also their imaginary parts evaluated at their pole frequencies:

(15.63)


Inserting this complex phonon frequencies into the formalism leads in the lowest approximation to the following refinement of the quasi-harmonic two-phonon Green function, (12.11):

G w _ _ 2(wi +(1)(1 +n 1 +n2 )

2,A1A2()- (w+i(Fl +F2 ))2_(wi +wl?

2(wi -wl)(n2 -n 1 ) (15.64)

Here the formal entity G which ensures the proper causal behavior of the freephonon retarded Green function is replaced by the sum of the line-widths of the phonons. In this way, a further step into a self-consistent treatment of the anharmonic self-energy with respect to all phonons involved has been done. The main approximation in (15.64) is the neglect of the imaginary parts of the occupation numbers i.e, correlated fluctuations of the occupation numbers during the absorption process.

FISCHER (1974) has discussed some qualitative consequences of this approach for the line-width and shift of the resonance frequency. He found, in complete analogy to the simple case of a local mode side-band (ZEYHER and BILZ (1967), see the discussion in Sect. 15 f) a smearing out of features in the selfenergy, in particular a 'filling up' of frequency regions with small damping from the neighboring regions with higher damping. This is exactly the case in the surrounding of the infrared resonance frequency and leads to an additional enhancement of the line-width near w R . Though quantitative calculations have still to be carried out the conclusion seems to be very plausible that quartic anharmonicity is often overestimated as was suggested by MOOIJ and E.R. COWLEY.

The foregoing discussion holds in principal also for the longitudinal optic mode wLO' The experimental basis for observing the properties of WLO is the thin-film technique first used by BERREMAN (1963). Physically, we may express the effect of a thin film by stating that the infrared resonance frequency wR with respect to the parallel component of the incident field is shifted to wLO' As a consequence, the self-energy of the resonance mode in such an experiment is just that of the bulk infrared frequency evaluated at the frequency WLO and with replacing wR by WLO:

(15.65)

Since usually WLO is very close to the maximum of the two-phonon damping function, the absorption is stronger near WLO (see Sect. 15 a). In addition, the damping function is most affected in the neighborhood of WLO by Coulomb anharmonicity, second-order dipole moments, etc. as was discussed in the foregoing sections. This makes a reliable quantitative analysis of IILO(w), though very interesting, at the same time extremely difficult. We refer the reader for details to the papers by LOWNDES (1970) (temperature dependency of WLO in alkali halides), E.R. COWLEY (1972) (self-energy of WLO in NaCI), HISANO et al. (1972) (analysis of Berreman experiments in NaCI and KBr), and BRUCE (1973) (self-energy of WLO in KBr).


The last point to mention in this section is the temperature dependence of the oscillator strength of the infrared resonance oscillator which is given by the integrated absorption:

(15.66)

On the other hand, the difference eo - eoo is related to the transverse effective charge Zt (6.48) so that the temperature and pressure dependence of the oscillator strength may be equivalently regarded as that of the transverse effective charge which describes the macroscopic dynamical dipole moment of the dispersion oscillator driven by an external field. As has been discussed in the foregoing paragraphs for the self-energy the change of eo - eoo with T and p is mainly a quasi-harmonic effect i.e. depends essentially on the change of crystal volume. Usually it is in the order of a few percent when going from room temperature to OK. The situation is different for highly polarizable or ferroelectric crystals where eo becomes a strongly temperature dependent function (see Sect. 17 e). In alkali halides a perturbative treatment of the temperature effect is satisfactory (SZIGETI, 1961; LOWNDES, 1970, 1972).

m) Discussion of other diatomic ionic crystals. In the foregoing sections we have seen that the infrared absorption of alkali halides is mainly described by an effective nearest-neighbor cubic anharmonicity aside from minor corrections due to Coulomb anharmonicity, quartic terms in the potential and non-linear dipole moments. If we proceed to other cubic ionic crystals, the situation does not change drastically as long as the polarizability of the ions is not very high.

oc) Silver halides. Of special interest are the ionic compounds of silver which exhibits unusual ionic properties, such as extreme diffusivity etc. In Sect. 4d we have compared the lattice vibrations of AgCI with those of RbCI. While the acoustic branches of both substances are still rather similar, there are two typical differences in the optical branches. Firstly, the transverse optic branch is lowered especially in the (111) direction caused by d-shell deform abilities of the Ag+ ion. Secondly, the frequencies in the longitudinal optic branch are increased since the smallness of the Ag+ ion leads to a contraction of the lattice and, subsequently, to an enhanced coupling between second-nearest neighbor chlorine ions.

The results of the lowering of the TO phonons is a corresponding decrease of the combination branch TO+TA at the L-point which governs the first maximum of the damping function in the cubic approximation (Sect. 15c). This maximum is situated in alkali halides near wLQ ~ 1.5 W R but is shifted in AgCl and AgBr to a frequency close to the infrared eigenfrequency wR • Therefore, the transmission bands in the silver halides are rather broad as compared with those of the alkali halides even at low temperatures (BOTTGER et aI., 1967).

Recently the anharmonic properties of silver halides in particular those of AgCl have been discussed by FISCHER (1974). He showed that an effective nearest-neighbor cubic anharmonicity is well able to describe consistently the thermal expansion, the temperature dependence of W R and, to a certain extent the infrared spectra in AgCI. Unfortunately. the measured reflection spectra


o 2 6 8 10 12 v(Tllz)--

Fig. 15.22. Frequency-dependent shift function LlR(w) for AgCl at 290 K calculated with two different widths B for the two-phonon states (FISCHER, 1974)

(HADNI et a!., 1968) seem to be unsatisfactory, probably due to surface and defect scattering.

An interesting aspect of Fischer's calculation is that he considered the influence of the phonon line-width in the two-phonon decay channels (15.64). As a consequence, the detailed structure in the phonon shift-function is washed out (Fig. 15.22). This might explain the fact that anharmonic renormalization of phonon frequencies as discussed by COWLEY and COWLEY (1966, 1967) does not lead to easily observable structures in the spectral functions i.e. neutron scattering profiles as one might expect from the above-mentioned calculations (see also MARIS, 1965; BILZ et al., 1974).

Among the silver halides, AgI has wurtzite structure but, under pressure, becomes again cubic. This crystal provides therefore a link between the crystals with sodium chloride and those with wurtzite (or zinc blende) structure. An investigation of its properties near the phase transition would be of great interest.

{3) Cubic oxides. The diatomic oxides show spectra similar to those obtained from alkali halides. The best investigated among these crystals is MgO. The optical constants have been measured by HAFELE (1963) and by JASPERSE et a!. (1965) while a theoretical discussion was given by BORIK (1970) (cf. Sect. 17).

One might expect that a partial covalency of these crystals could lead to an enhancement of the relative importance of the non-linear dipole moments. On the other hand, the increase of the ionic charge from one to two supports the absorption mechanism via the Reststrahlen oscillator so that an effect of the second order dipole moment cannot be evaluated from the experimental spectra (BORIK, 1970). In this respect, the cubic oxides show a behavior very similar to those of the alkali halides. It is interesting to note that the situation in Raman scattering is quite different where the unusual polarization properties of the oxygen ion dominates the situation (see Sect. 18).

y) Alkali halides with cesium chloride structure. It was shown by MAHLER and ENGELHARDT (1971) that the lattice dynamics of the alkali halides with cesium chloride structur, CsCI, CsBr, and CsI can be calculated in a very similar way to that of the other alkali halides by using the breathing-shell model (SCHRO-


DER, 1966). Consequently, they tried to calculate the infrared absorption of these crystals along the lines which have been followed in the foregoing sections. The optical data of CsBr have been measured at room temperature by GEICK (1961). MAHLER and ENGELHARDT evaluated the experimental data with the help of (15.42) and (15.49) and obtained empirical shift and damping functions. They compared this with a theoretical calculation where only rti3 )(w) (15.6) was determined in an effective nearest-neighbor approximation. Since the symmetry is different from that of the sodium chloride structure, the nearest neighbor central potential (Born-Mayer type) exhibits a different analytical form for the phase tensor K3 and the eigenvector tensor f3. An important point to note is that the selection rules forbid two-phonon processes at the points r, X, K and L (BIRMAN, Table 46. b, p.421, and Fig. 19, p.420).

The results of an one-parameter calculation for the damping function by MAHLER and ENGELHARDT are shown in Fig. 15.23. The agreement with the empirical damping function is quite good if one considers the relatively high temperature and the many approximations. The same is true for the comparison between empirical and calculated shift function as shown in Fig. 15.24. The main peaks originate from TA + TO and TA + LO combinations as in the other alkali halides but they cannot be traced back to critical point combinations as in the former case. - It should be noted that the second order dipole moment may play a more important role as in the other alkali halides. The application of Borik's criterion, (15.37), leads to values for W*/WR between 2 and 3. Further measurements at low temperatures are necessary to clarify this point.

6) Lead chalkogenides. (For a review see DALVEN, 1969.) Among the highly polarizable diatomic cubic crystals the compounds PbS, PbSe and PbTe are of special interest due to their nearly ferroelectric behavior (80 =412 in PbTe) and the effect of free carriers on their properties. A detailed study of the dynamical properties of PbTe has been given by COCHRAN et a1. (1966). The main feature is the strong dispersion of all branches in particular that of the TO branch. The TO mode at r is not very sensitive to the amount of free carriers (weak plasmon-phonon coupling, see COCHRAN et aI., 1966; VARGA, 1965).

Since the best-fit shell model contains 15 parameters (static ionic charge IZI ~ 2) one may not expect a simple description of the anharmonic properties. It seems, however, (COCHRAN et a1.) that an effective rigid ion model (IZI ~ 1) with a strong nearest neighbor cubic anharmonic coupling is able to give a satisfactory description of the lifetime of the transverse optic mode wR .

Recently, BURKARD et al. (1974) have analysed the infrared spectra of PbSe along the lines pointed out in Sect. 151. Their results for the damping function of PbSe as determined from the reflection spectra (Fig. 15.25) are shown in Fig. 15.26. The essential difference as compared with the alkali halides is the fact that the infrared resonance frequency is very low and therefore imbedded in the high-density region of difference processes. This leads to a much stronger variation of rR (w) in the reflection spectra. Aside from this, the analysis is not essential different from that of the alkali halides if the very low frequency regime is corrected for a contribution of the free carriers by a Drude-term. An analogous problem appears when analysing the low-gap III - V compounds, like InSb (refer to Sect. 17).

Sect. 15

t

c


z 1111 1111 II II

+5tekhanov

tNarayanan

3 4-6)(10135-7;-

III I I II I III I I II I III I I II I

b R 1----------t-lt--!--h-lt1I-'-HIt---------j III

x~---------~--~~~~~~--------~

t

a

201

Fig. 15.23a-c. Analysis of the damping function for CsBr at 300 K. (a) Calculated: stepped curve, evaluated from experimental data: smooth curve; (b) sum dispersion curves; (c) two-phonon den

sities, (15.10) and Raman measurements (arrows). (MAHLER and ENGELHARDT, 1971)


7~------~~------~--------~-=~~r-~--------~--'

Fig. 15.24. Shift function for C sBr at 300 K calculated (stepped curve) and evaluated from experimental data (smooth curve). (MAHLER and ENGELHARD, 1971)

IJItilJ :rm:t -J

:llTIldd o 50 700 750 200 250

w(cm-1) --

Fig. 15.25. Reflection spectra of PbSe at three different temperatures. -- Measurements; --oscillator fit plus Drude term. (BURKHARD et al., 1974)

For a recent discussion of the dynamical properties of lead chalkogenides see BUSSMANN-HoLDER et al. (1983).

£) The discussion of the four different groups of diatomic cubic crystals in the foregoing paragraphs may be regarded as representative for the whole family. It is hoped that the discussion has pointed out the important features for the understanding of the infrared spectra of these crystals. It seems that the (effective) short-range cubic anharmonicity is governing the situation in strongly ionic crystals but the behavior of the more strongly polarizable crystals still


300K

r 250 I--\,.--J-~--I---t-----t

o 50 700 750 2Da w(cm-1)_

Fig. 15.26. Imaginary part of the dielectric constant for PbSe at three different temperatures (BURKHARD et aI., 1974) -- Kramers-Kronig analysis; --- oscillator fit plus Drude term; -.-.-

Drude term; ...... imaginary part of B- 1; -- infrared oscillator

needs further investigations. For further informations on the experimental spectra and their theoretical analysis the reader is referred to the infrared bibliography by PAWLIK (see MOLLER and ROTHSCHILD, 1971).

n) Cubic crystals with three and more ions in a cell. The cubic crystals possess a diagonal dielectric tensor a(w) due to their high symmetry. The general structure of a(w) for a crystal with many infrared-active dispersion oscillators was discussed in the harmonic approximation in Sect. 12. Equation (6.67), gives a general tool to determine the different transverse and longitudinal optic frequencies, W Ti and WLi ' as poles and zeros from this equation. The problem is to extend this equation when considering the finite linewidth of the optic modes.

(J() Crystals with one dispersion oscillator (CaF2' etc.). We can first separate out from the further discussion the cubic crystals with more than two ions in a cell but possessing only one infrared-active mode. The prototype of this family of crystals is CaF2 • (The vibrational spectra of this compound and its homologues are shown in BILZ and KRESS (1979).) The important point to note is that the additional three optic modes as compared to the diatomic crystals are degenerated at the r point in the B.Z., i.e. they have zero oscillator strength. (Their importance for the Raman spectra is discussed in Sect. 18.) This is correlated with the even parity of these modes: the calcium lattice is at rest while the two fluoride sublattices move in a symmetrical way against each other. There is therefore no direct coupling between these modes and the transverse optic modes and their main effect is to contribute to the damping function by allowing for further two-phonon combination bands. One may still expect that the first maximum of the damping function is close to TA + TO as in the alkali


halides and that no qualitatively new features appear in the spectra. The first analysis of the infrared spectra of CaF2, SrF2 and BaF2 was given by W. KAISER et al. (1962). They fitted two classical oscillators to the experimental data:

(15.67)

where the frequencies of one oscillator are always close to the resonance frequencies WR,i while the frequencies of the second 'oscillator' turn out to be close to the frequencies of the inactive Raman oscillators at q = 0 which have been identified much later by neutron spectroscopy. This seems to be a similar accidence as the co-incidence of WLO and the two-phonon maximum in the alkali halides. The procedure of fitting a series of classical oscillators to the measured spectra is quite common in the literature but, while providing a formal description of the spectra, very unsatisfactory from a physical point of view as was mentioned by KAISER et al. All three spectra show a rather sharp peak near the end of the two-phonon spectra (Fig. 15.27). By inspection of the dispersion curves they may be clearly identified by TO + LO combinations. The essentially exponential decrease of oc(w) with increasing frequency is clearly recognizable from Fig. 15.27.

For the other cubic crystals with only one dispersion oscillator the reader is again referred to the bibliography by PALIK (see MOLLER and ROTHSCHILD, 1971) and the current literature.

/3) Crystals with several dispersion oscillators. The problem of crystals with more than one active vibration was mentioned in Sect. 12 and a formula for the case of two coupled oscillators was given in (12.42). We follow here the discussion of this problem in some detail to clarify some general points in the case of coupled active modes.

In the harmonic limit (Yi~O) (15.67) is equivalent with (7.112), if the oscillator strengths !;o=4npi and the longitudinal frequencies wLi satisfy the following equations:

=s (W£l-W2)(W£2- W2) 00 (wil -w2)(wi2 -w2)

and (generalized Lyddane-Sachs-Teller-relation)

This leads to following quadratic equation for the wLi's:

(15.68)

(15.69)

(15.70)

Sect. 15

i ~ U

~

~

200

100

60 40

20

10

6 4

2

1.0

0.6 0.4


~~ --

~. ;);> Ca F2

~ tf ~3000K 77° K'\ ~

~ u.. t\". "'-

, ..... ~ ~ ~ ~

""-Y...

550 600 650 700 750 800 850 900 950 10001050 1100

a P IN CM-1

200

~ SrF2 100

60 40

20 "i ~ 10 u

'" .~ "" )U'"

r\w /' ~ \ ~

~ 6 ~ 4

2

""'r,. '""'q,

77!'K, I~O(j'K

" " 0.6 OA

........ '" '" ~

500 550 600 650 700 750 800 850 900 950 1000 b YIN CM-I

30

200

0

i~ 0 10

8 6 0,\ '0", o ~

40

I 2 ~

0

u 0 8

~ 1

~ 6

4

2

0 8 6

1. O. O. 0.4

¢. 'l}

boa:t"

~

BaFz

.~

~~

~ ~ ~. ~ 3000K

'''' " ~ "-77°K' ~ , " P-...

"- ~

r... 'L rt..

"n

"" 450 500 550 600 650 700 750 800 850 900 C PIN CM-I

205

Fig. 15.27. (a) Absorption coefficient of CaF 2 at 300 and 77 K; (b) absorption coefficient of SrF 2 at 300 and 77 K; (c) absorption coefficient of BaF 2 at 300 and 77 K. (W. KAISER et aI., 1962)


since

(15.71)

In the case of weak oscillator strengths, J;~Boo' (15.70) reduces to

2 Boo + 11 2 WL1~---WT1

Boo

2 Boo + 12 2 WL2~---WT2'

Boo

(15.72)

which means that we still obtain approximately separate Lyddane-Sachs-Tellerrelations for the single oscillators while in the strong coupling case, 11 ~ 12 ~ Boo'

wt2~wi1 +Wi2 (15.73)

2 ~ 11 2 12 2 ( 5 74) WL2~-WT1 +-wT2 1 . Boo Boo

i.e. a strong constructive interference between both oscillators. We try now to proceed in an analogous way in the case of coupled

anharmonic oscillators. Starting from (12.42) we obtain:

with

and

2wTl (wT2 _w2) G2(w) G1(w) (15.75a)

(Wi1(W)-w2)(wi2(W)-W2) G2(w)-4P12 WTl WT2

P12 = L112 L121 - r12 r21 - i(r12 L1 21 + r21 L1 12),

G12(W)= -GR, 1 (L1 12 -irdG2 (15.75b)

and analogous equations for G21 and G2 in a self-explaining short notation. If we now regard the case of weak anharmonic coupling, II12~II1,II2' we obtain with (12.3) and consequently neglecting P12'

L - WI - W 2 Bo-Boo= 4n X (0)=/ITG1+/2W2TG2 (15.76)

at the generalized Kurosawa-formula (7.148) (BILZ and GENZEL, 1974)

(wL -W2)(wt2 _w2) Bo=Boo (-2 2)(-2 2) (15.77) wTl-w wT2 -w

with

wL,2(W)=2! [(Boo + J1)wL(w)+(B oo + J2)wt2(W) 00

± {(Boo + Jl) Wil (W)-(Boo + J2) Wi2(W)+ 4Jd2 Wi1 (W)Wi2(W)}] (15.78)


and

(Wo)2 ];= /; W: . (15.79)

In the case of small oscillator strengths, 11;1 ~soo' we obtain analogously to (15.72)

(15.80)

In this weak coupling limit we can therefore replace the harmonic resonance frequencies W Li and W Ti by their pseudo-harmonic equivalents WLi and wTi . The important point to note is that the frequency dependence of the imaginary part of these resonances is essentially the same aside from a'Lyddane-Sachs-Teller'factor (soo + ];)/soo' This result is in complete agreement with that of the single anharmonic oscillator case. On the other hand, (15.77) contradicts the treatment often used in the present literature, where the damping function of the different resonances is always taken at the individual pole frequencies. This error becomes very important if resonance frequencies are not well separated because then one has to consider that the damping functions, when going from a specific wTi to the corresponding W Li may change by an order of magnitude.

From the general structure of the proof of (15.77) it can easily by shown that in the weak coupling limit a generalized Kurosawa formula for a system with several oscillators is obtained:

n

TI (wL-w2 )

s(w)=soo _i:_l ___ _ (15.81)

TI (wi i -W2) i= 1

With the additional assumption of weak oscillator strength, 11;1 ~Soo' one obtains again

(15.82)

This formula may be useful for a qualitative discussion of crystals with many infrared active dispersion oscillators.

In the strong coupling limit where II21 may be comparable with the IIi' the situation becomes much more complicated since then, in (15.73) and (15.74) the harmonic frequencies W£i have to be replaced by strongly frequency dependent renormalized frequencies wL. In addition, the coupling may become the dominating effect in the coupled system. This seems to hold in the case of ferroelectrics with cubic symmetry. Among the cubic ferroelectric crystals (Oh symmetry) with several dispersion oscillators the most interesting are the oxidic perovskites. The main feature of these substances is the appearance of a soft transverse optic mode at q ~ 0 which is correlated with strong anharmonic interactions between the ions. We shall discuss the infrared properties of the perovskites in some detail in Sect. 17 e.


16. Infrared spectra of covalent crystals.

a) General features of the spectra. In the following we discuss the infra-red absorption in centro-symmetric crystals with equal lattice constituents, such as the rare-gas crystals and diamond with its homologues silicon, germanium and grey tin. This crystals do not possess any infra-red active oscillator due to inversion symmetry so that no resonance-absorption is observed. The only possible absorption by phonons is through higher order processes where the deformable electronic charge density contributes to the dipole moment. This decays through non-linear electron-phonon potentials into two or more phonons and leads so to effective non-linear dipole moments. The corresponding diagram is shown in Table 15.3 (diagram a).

The general theory of the absorption in this case is described in Sect. 12. The effect of the non-linear dipole moments on the absorption in ionic crystals is given in Sect. 15 i.

The dielectric constant is

(16.1 )

where the susceptibilities lZJ and X[3 J for second- and third-order dipole moments are given by (12.19). The real part of the is is so small that its contribution to e'(w) can be neglected, an estimation of this contribution with the help of the Kramers-Kronig relations, (7.49), for silicon or germanium shows that it is less than 0.1 % of Coo at room temperature. To a very good approximation we may therefore use the following equations

(16.2)

4n 1 '\' '\' - -e"(w) =3 [2 L.. L.. M2(.AIAZ)·M2(AIA2)

+.-A1A2

· 2(Wl±W2)6(w2_(Wl±wZ?)[nZ+t±(nl +t)J

+i L (2+1) L M3(AIAzA3)·M3(,{1,{Z,{3)

· 2(Wl +wz ±w3)6(wZ -(w 1 +W2±w3f)

· [(1 + n 1 + nZ)(n3 +t ± tH n 1 nz]' (16.3)

Equations (16.2) and (16.3) are reminiscent of (15.29) and (15.30) for the infrared absorption at very long wavelengths. In fact, this result has to be expected since the absorption under consideration describes that of an electronic oscillator at low frequencies, where the real part is given here by the electronic dielectric susceptibility Coo (instead of co) and where c"(w) is proportional to a damping function rND(W) due to non-linear dipole moments (cf. (15.28)):

(16.4)

Here, Coo -1 is the electronic part of the oscillator strength, and the frequency WeI corresponds to an effective energy band gap. The strength of the absorption is determined by the coupling parameters Mn which depend on the polarizability and the non-linear electron-phonon coupling.

Sect. 16 Infrared spectra of covalent crystals 209

The advantage for the analysis in this case is that no resonance effect obscures the details of the spectra as it does in the ionic crystals. Unfortunately, the intensities are rather weak. In the case of rare gas crystals the polarizability of the ions is so weak that experimental spectra have not been observed, so far. The only information at present is obtained from impurity-induced one-phonon infrared absorption (see Sect. 20b).

b) Spectra of crystals with diamond structure. The spectra of Ge, Si and C have been measured by several people (Ge: FRAY et aI., 1965; Si: JOHNSON and COCHRAN, 1962; BALKANSKI and NUSIMOVICI, 1964; C: ANGRESS and SMITH, 1965; GOODWIN et aI., 1967). The simplest possible interpretation is done again by a critical point analysis which has been discussed in Sect. 15 c for ionic crystals. This analysis is based on the selection rules for two- and three-phonon processes which are presented in detail in Birman's article in volume XXV /2 b of this Encyclopedia and summarized in Tables 40.7 and 40.8. The identification is supported by the homologuous behavior of Ge and Si (Fig. 15 in Birman's article, p. 416). Results have been given for Ge and Si by BALKANSKI et aI. (1963), BILZ et aI. (1963), and JOHNSON and LOUDON (1964). The critical point analysis is not completely free of ambiguities and possible errors. The reason lies mainly in rather small deviations of the lattice dynamical model used in the calculations from the actual dispersion curves. This leads to uncertainties in the analysis if many critical points come together in a small frequency regime.

An improved understanding of the spectra is obtained by a dynamical treatment of the coupling parameters. They were discussed first by BORN and HUANG (1954) and LAX and BURSTEIN (1955). WEHNER (1966b) has shown that a nearest-neighbor approximation to the coupling parameters (STEPHEN, 1958; DOLLING and COWLEY, 1966) of the second order dipole moment is incorrect with respect to infinitesimal translation and rotational invariance. The physical reason is (SZIGETI, 1963) that in a crystal consisting of physically identical atoms and possessing an inversion center between every pair of atoms the coefficients M(LI;) cannot be derived from two-atom interactions. This leads to secondnearest neighbor interactions and at least seven free parameters for the M's (WEHNER, 1966b).

With the help of these parameters WEHNER, BORIK and KRESS analysed the infrared spectra of diamond (WEHNER et aI., 1967) and those of Si and Ge (KRESS et aI., 1968). The coupling parameters have the following structure:

M 2(AX') = const. (w",w);,)-1/2 . Mv

. K~o(q) . [e(K/ A) e* (K'/X') - e* (K"/X') e(K'1I / A)] (16.5)

where the eigenvectors e and the phase K3 exhibit the similar structure of M 2 as compared with V3 , (15.5) to (15.7). The discussion of the seven parameters Mv and of K3 is given in the papers by WEHNER (1966b) and KRESS et aI. (1968).

The results for diamond are shown in Fig. 16.1. Only five parameters turn out to be relevant and give a very good fit. An interesting result of this analysis is that the inclusion of the anti-symmetric second nearest-neighbor force con-


["

12 DIAMOND J f22' 4'4 b 01.5

lW II 10 A=

B=

8 c= 0=

£ 6 E= .C! :;, F= ~ '" ~

4 .Q

~ 2

l~ Itl d -b-I.5

14 ···········/,-00

12

10

8

6

4

2

E" A= 10 0.45

B= 0.28

8 c= 0.22

0= 0.42

~ 6 E= 0.06

§ F= 0.42

~ 4 G= -0.05

e "" -e 2 e.

2.6 G)(/O'4 sec-I)

Fig. 16.1 a-c. Diamond, room temperature. (a) Two-phonon infrared absorption, experimental (continuous curve) and theoretical (stepped) with i5 = 1.5. Dipole moment parameters A to G in units of elementary charge per lattice constant; (b) two-phonon density of states; (c) two-phonon

absorption i5 = O. (WEHNER et a!., 1967)

stant (j which only affects the }; branches in the description of the dispersion curves of diamond leads to a serious change in the two-phonon density of states.

Analogous results for silicon are shown in Fig. 16.2. In this case, a good fit with 3-4 parameters of the second order dipole moment may be obtained. We note that the results by DOLLING and COWLEY (1966) for the infrared spectra of C and Si are partially incorrect (refer to KRESS et aI., 1968).


50 Si 29] K

40

]1 ~f5:3 30 ",,-"

"'I::,. 20

10

a 0 0.6 0.8 16 18 20

t 200 41~ 0.0 A2~ 0.98

-::S-150 A3- 0.68 ~ A.- 0.03 ~ ~ 100 A5--D.18 ~ 46= 0.58 ~ A7 =-0.01 ,

50 '"

U 161820 b CJ(101"s-~-

Fig. 16.2 a-b. Silicon 293 K. (a) Two-phonon density of states; (b) measured (solid line) and calculated (histogram) infrared absorption. Dipole moment parameters Al and A2 in units of elementary

charge per lattice constant. (KRESS et ai., 1968)

Si 29J K ! 2°9-_....:5:::+6~ __ + __ -O ________ _

~1;3' 6:

4+54+6

15 3+6 3+5

lOr-~=::::::-.t:::tQ--.--=:,..."..=i'~~,q.i-=~:~·C~- ~-I --~-..... _________ C<-: ...... \ Or

1+2 ...... )

0.5

°W·~~-LK~~LL~K~LL~W~-------------~~W~~x~KLU~~r

(I-M+f,O) (!-&i-Us) (J+lf,f-s) (t,}io) (1-f,1-fO) ~-c"(arbifrory IJnits)

Fig. 16.3. Silicon, 293 K. Measured (solid line) and calculated (dotted line) infrared absorption together with the dispersion curves of summation branches. Fobidden combinations: circles.

(KRESS et ai., 1968)


The calculation has been compared by the authors with a critical point analysis from a shell model, which is shown in Fig. 16.3. A comparison with an earlier analysis confirms many of the critical point assignments but gives some further evidence for the fact that maxima in multiphonon spectra are sometimes due to pure density effects not related to critical points (BILZ et at, 1963).

The calculations of the same authors for Ge are less satisfactory as can be seen by inspection of Fig. 16.4. The fitted parameters of the second order dipole moment do not exhibit clear relations to those obtained for silicon. In addition, the eleven parameter shell model (DOLLING and COWLEY, 1966) used in this calculation shows some deviations from the experimentally identified peaks. The difficulties of critical point assignments to experimental peaks, when based on a partially inaccurate phonon model, are quite obvious in this case.

These investigations show that a few parameter are sufficient for the analysis of the infrared spectra of diamond and its homologues and give satisfactory results in the two-phonon regime. The three-phonon spectra of these crystals which are experimentally clearly identified have only given a qualitative critical point analysis (BILZ et at, 1963).

6e 77 K

ao to t2

I (A1014S-~-

"?;- 20 A1 = a01 ~ Az=aOl §

~ AJ -!l/J9 it! A~-a27 '§

]. 10 A5=020 :.. A6 -f1IJ4

A7 = all

b 0 a6 12

Fig. 16.4a, b. Germanium, 77 K. (a) Two phonon density of states; (b) measured (solid line) and calculated (histogram) infrared absorption. Dipole moment parameters At to A7 as in foregoing

figures. (KRESS et aI., 1968)


A deeper insight into the origin of the coupling parameters should be obtained from a treatment which uses a plausible model potential for the nonlinear electron-phonon interaction in covalent crystals. It seems that the shell model with forces acting through the shells is not an appropriate starting point for performing such an investigation since its concept of well separated and weakly polarizable ions is not in accordance with the properties of the electronic charge density in these crystals. Apparently, the great number of parameters used in shell model calculations for germanium, etc. supports this criticism. A more realistic attempt for a determination of infrared coupling parameters may start from models which take explicitly into account the 'bond charge' between the ions which is connected with the highly directional properties of the interatomic forces.

An attempt in this direction has been undertaken very recently by Go et al. (1974). They used the adiabatic bond-charge model which was developed by WEBER (1973, 1974) (see Sect. 5 d). A non-linear extension of this four-parameter model leads to a short-range potential of the type V; (see Table 15.4) which describes the cubic coupling between the electronic bond charge and its nearest neighboring ions in the elementary cell. The potential corresponds formally to four independent coupling parameters but, practically, the assumption of a central potential for the relative displacement of the bond charge against a neighboring ion (with only two independent parameters) satisfies the experimental data quite well (Fig. 16.5).

The success of the adiabatic bond charge model exhibits an important analogy to the situation in ionic crystals if regarded from the point of view of the microscopic theory. It demonstrates the basic difference between infrared

"iii 15

C ::J

Ge T= 77K

-- Theory, ot = 1 , fJ= 0, Y= Q 154

---- Exp.

(J) (x 1014 5-1 )

Fig. 16.5. Germanium 77 K. Measured (dashed line) and calculated (solid line) infrared absorption. Bond-charge model parameters: 1X=1.0, /3=0.0, ]1= -0,16 refer to WEBER et aI., 1975


absorption and Raman scattering in insulators. As was discussed in Sect. 6, the stability of the insulating crystals depends on the off-diagonal elements of the dielectric function which ensure the validity of the acoustic sum rule. - In ionic crystals, the rigid-ion model shows all the important features of the situation, where the infrared absorption originates essentially from the cubic 'off-diagonal' interactions between the weakly overlapping 'static' charges. - In covalent crystals, the off-diagonal part of the dielectric function may be represented by the electronic bond charge (R. MARTIN, 1969). Again, a non-linear extension leads to the description of the non-linear moments in infrared absorption. In both cases, the model works particularly well in the limit of weakly polarizable crystals (e.g. LiF or diamond) i.e. if the diagonal part of the dielectric function (represented by Goo at long wavelengths) is rather small. It is this diagonal part, however, which determines the polarizability and, in a non-linear extension, the Raman scattering of the crystal. While the shell model provides the necessary step beyond the rigid-ion model in ionic crystals in order to describe the polarizability and the Raman effect (Sect. 18) we need an analogous extension of the bond-charge model with a proper representation of the diagonal part of the dielectric function (R. MARTIN, 1969) in order to understand the Raman scattering in these crystals (Sect. 19).

c) Covalent crystals with linear dipole moments. Polyatomic crystals with equal lattice constituents but lacking a center of symmetry may exhibit a linear moment connected with one or more optical phonons with very long wavelengths (q~O). Typical examples of such crystals are selenium, tellurium and sulfur. Since there are three particles in a cell one obtains two sets with three optical branches for each. Some of them are infrared active (in spite of the fact that the atoms are neutral in the equilibrium position) due to the anisotropy of the charge distribution in a displaced ionic configuration.

The lack of inversion symmetry means that the dielectric constant becomes a tensor. It is therefore more practical to discuss the infrared spectra of these crystals together with those of the other non-cubic crystal (Sect. 17b).

17. Infrared spectra of crystals with mixed ionic and covalent character

a) The concurrence of anharmonicity and non-linear dipole moments. In this section we discuss crystals where the anharmonic coupling between phonons and non-linear dipole moments are of equal importance. We remember that in strongly ionic crystals like the alkali halides the effect of non-linear dipole moments was rather small (Sect. 15 i). In connection with this problem the theoretical formalism was completely developed and it can be used here.

The interesting point to note is, that there exists no continuous transition from purely covalent to partially ionic crystals (except, perhaps, in mixed crystals) from the point of view of infrared absorption. We may, for example, consider GaAs as a compound still very close to silicon, because both substances show very similar phonon spectra and possess rather similar electronic band structures. This similarity shows up in some quantitative features (similarity of phonon-combination frequencies etc.) but they do not affect the great differences with respect to infrared absorption.

Sect. 17 Infrared spectra of crystals with mixed ionic and covalent character 215

Firstly the dynamical 'ionicity' of GaAs is strong enough to make the resonance part of the absorption as important as the part due to non-linear dipole moments. This happens in spite of the strong increase of this latter part due to the contribution of two-body (Ga- As) forces which are forbidden by symmetry in silicon (Sect. 16b). The problem was discussed by SZIGETI (1963) and GEICK (1963). The breaking of the exchange symmetry between neighboring particles in the III -V compounds constitutes, therefore, a gap between the dynamical theory of coupling parameters in both cases. This is reflected, for example, in the bond charge model by a sudden increase of the number of parameters if one tries to transfer and to extend the calculation of the non-linear dipole moment in Si and Ge to the case of GaAs and other III -V compounds. It was discussed in Sect. 16 that the adiabatic bond charge seems to play the role of an effective electronic transverse charge in the infrared absorption of the pure covalent materials (Go et aI., 1975). It was shown by RUSTAGI and WEBER (1976) that the lattice vibrations of GaAs may be obtained from those of Ge mainly by shifting the bond charge a little bit from its midpoint position in a bond in the direction of the neighbouring As ion. This parallels the well-known shift of the charge-density maximum in GaAs as compared to Si (WALTER and COHEN, 1971) and lends support to the idea that the bond charge is some model representation of the overlap portion of the electronic charge density centered at the charge-density maximum. If we, in a first approximation, replace the shift of bond charge by keeping it at its original position but adding an appropriate bond dipole we see immediately that the ionic effective transverse charge Zt which is related to the infrared reflectivity should be strongly related to those effective bond dipoles between the neighbouring ions and to their (new) interactions with them. In this picture it makes no sense to connect the effective charges Zt with the static charges Z of the single ions since the main effect lies in the strong deformation of the electronic charge density and its asymmetry with respect to neighbouring ions (BURSTEIN, 1965; SZIGETI, 1973). It would be interesting to see a calculation with a nonlinear bond charge model where both effects that of the electronic bond charges leading to non-linear moments and that of bond dipoles responsible for the occurence of effective "ionic" charges are considered.

Another possibility is the extension of the non-linear shell model used in Sect. 15 for the case of weakly ionic crystals. This corresponds to a multipole expansion of all interactions at the ionic lattice sites and describes the situation in terms of "ionic" charges, polarizabilities, etc., and the corresponding Coulomb and short-range interactions. The success of such a description means essentially that the multipole expansion is rapidly converging, perhaps due to some lucky cancellation of different contributions and more probably because of the strong screening of long-range forces in these highly polarizable materials (refer to Brout's sum rule, Sect. 5.f). It does not mean, of course, that the ionic quantities as determined such a model have the same significance as the corresponding ones in strongly ionic crystals where they are close to the free ion values. The evidence for that may be found, for example, in Table 6.1 which shows that the difference between the transverse effective charges (about 2) and the Szigeti effective charges (about 0.5) is very large in the II-VI compounds. We


may therefore expect that the infrared absorption due to an ionic dipole moment related to Z:j: and an electronic (non-linear) dipole moment are of equal importance (refer to (15.35)).

A similar situation occurs in covalent systems such as Se and Te where instead of the charge asymmetry an equivalent anisotropy leads to the appearance of anisotropic effective charges. The differences between transverse and Szigeti charges are again very large (refer to Table 6.1).

The situation is quite different in the II-VI compounds. Here, the bond-charge model does not work, at least not in its simple form (RUSTAGI and WEBER, 1976), since the asymmetry of the bond charge position becomes so large that it is instable against displacements from the new equilibrium position. As a consequence the four bond charges re-organize themselves around a group VI ion to form a negative ion with a nearly filled electronic (n p)-shell. This leads in a natural way to a description in terms of a shell model with strongly polarizable negative ions and some overlap polarizability of the positive ions (refer to Sect. 4).

Although the II-VI compounds, even those with NaCI-structure like CaO, still exhibit some covalency in their Raman scattering properties (Sect. 18) it seems that the description of their phonons with 5 or 6 parameters in an overlap shell model (BILZ et aI., 1975) is well satisfied. It should be noted that a bondcharge model calculation for MgO (GILLIS, 1971) including an asymmetric bond charge position and second nearest neighbor 0 - 0 forces but using Martin's restriction of keeping the bond charge on the ion-connecting line was quite successful in describing dispersion curves. It would, of course, not provide a basis for the calculation of Raman spectra.

b) Spectra of crystals with zincblende structure. IX) GaAs and GaP. One of the best investigated III-V compounds is gallium asenid. In Fig. 17.1 a comparison of the imaginary part of the dielectric constant in GaAs with that in Ge is shown (GEICK, 1965). Following Szigeti's (1963) suggestion that a considerable amount of the absorption in GaAs should be due to the nonlinear dipole moment, GEICK was able to show that on the average this contribution has nearly the same order of magnitude as the effect of anharmonicity in connection with the transverse effective charge Z:j: of GaAs.

A more quantitative analysis has been given by BORIK (1970). In terms of his effective frequency w*, (15.36), which turned out to be ~0.25. He found that the second order dipole absorption process becomes dominant for frequencies larger than 1.3 W R (Fig. 17.2) which is obvious four the sudden flattening of the e"(W) curve in this area. BORIK noted some clear discrepancies between his calculations and the experimental data because of deviations of the phonon shell model of DOLLING and WAUGH (1963) from the experimental data. He found it impossible to give an analysis of the infrared absorption in GaP (KLEINMAN and SPITZER, 1960; CHEN et aI., 1966; FREY et aI., 1969) by using the shell model of YARNELL et ai. (1968).

P) InSb and InAs. The two-phonon infrared spectra in undoped InSb has been investigated by KOTELES et ai. (1974). They where able to correlate the frequencies and intensities of the two-phonon absorption spectra with calculated


/.2 15 2.0 25 3.0

Q - ~/w. --+

Fig. 17.1. Imaginary part of the dielectric constant en as a function of reduced frequency w/wR in GaAs and Ge. (GEICK, 1965)

20 Wjw.-

Fig. 17.2. Infrared absorption in GaAs (BORIK, 1970). Experimental values are taken from

two-phonon density of-states curves (Fig. 17.3). They found that the early four'effective'-phonons analysis by FRAY et al. (1960) was still useful for the interpretation of the spectra. An experimental determination of the optical constants of InSb with the help of an infrared interferometer is shown in Fig. 17.4.

218

(fl zf::: OZ i==> 0..>a:: a:: o<t (fla:: mt-<tiij

a:: <t

Interpretation of experimental spectra Sect. 17

W(cm-')

Fig. 17.3. Infrared absorption in InSb. (KOTELES et ai., 1974)

-If- InSb

1.0 :;;: >. 0.8 ·s ti 0.6 ~ ~ 0.4

0.2

0.0 0 300

10.0

5D

2D

,,' 1.0

0.5

0.2 0 300

wavenumbers (cm-1)

Fig. 17.4. Optical constants in InSb. (GAST and GENZEL, 1973)

Since InSb has to very small energy gap, the conduction band may be easily filled up even without doping as a function of temperature. This leads to a characteristic Drude part in infrared absorption of a strength which is characterized by a plasma frequency w;oc4nne 2/m (about 60 cm- 1 in Fig. 17.4) which exhibits a plasma edge shifting to higher frequencies with increasing density n of


electronic carriers. - Very similar optical constants are obtained for InAs (GAST, 1972).

y) Zinc-chalcogenides. The II-VI compounds with zincblende structure exhibit infrared properties which are much closer to those of purely ionic crystals. In Fig. 17.5 the room temperature reflection spectra have been analyzed in terms of a real and an imaginary part of the wave-vector given by:

q' =2nnw and q" = 2n kw. (17.1 )

They are obtained by fitting a classical dispersion formula to the reflection spectrum and analysing it with the help of a Kramers-Kronig transformation. Investigations of this type have been carried out by BALKANSKI et al. (refer to BALKANSKI, 1973) and HADNI et al. (1968). No detailed analysis of a damping function similar to those of Borik for GaAs has been given so far.

In Table 6.1 effective charges are collected for crystals with various structures which show the systematic increase of Z1 when going from III -V to II - VI compounds.

For further information on infrared spectra in crystals with zincblende structure the reader is refered to the review by SPITZER (1967) on multiphonon lattice absorption.

c) Spectra of perovskites. The cubic ABX 3 perovskites and their distorted modifications have been investigated in detail during the last decade. Among them the oxides AB0 3 with strong para- or ferroelectic behavior attracted particular attention. (For general reviews of the infrared dielectric properties the reader is refered to BARKER, 1969, PERRY, 1971, and LINES and GLASS (1977)).

The stimulation of these investigations came from Frohlich's and Cochran's discussion of the importance of soft modes for the interrelation of crystal stability and ferro electricity (FROHLICH 1949; COCHRAN 1960a, b). This shows up most clearly in the temperature dependence of the static, dielectric constant in the para electric regime:

C 8 0(T)=-

T- T" (17.2)

where Tc is the Curie-WeiB transition temperature. The very high values of 80

dominate the long-wavelength infrared spectra at low temperatures. To understand the situation more precisely we discuss in detail the case of

SrTi03 which is the best investigated of the ferroelectric perovskites and is, in many respects, similar to BaTi0 3, KTa0 3 and other crystals. The crystal structure is shown in Fig. 17.6. It is reminiscent to the simple diatomic rocksalt structure (Ti - ° octahedra, Sr - ° planes) and cesium chloride structure (Sr - Ti0 3) which facilitates a qualitative analysis of the forces and dynamical properties of the crystal. The phonon dispersion curves have been measured by STIRLING (1972, 1976) and analysed in terms of different shell models. A few of his results in the main symmetry directions are shown in Fig. 17.7. It is useful to compare these dispersion curves with those in SrO (MIGONI et aI., 1975), Ti02

(TRA YLOR et aI., 1971), and TiN (Kress et aI., 1977). This shows that the very

220

.. E u

:u .0

E " c .. > 0

~

Interpretation of experimental spectra

R

90 10)

80

70

60

~ SO

~ 40 >

u

" 30

" 0::

20

10

1 SO 200 250 300 Wave number lem-1)

Ib)

200

Imaginary part of the propagation vector

q"= 2TtKw lem- I)

Real part of the propagatIon vector

q'= 2Ttnw lem- I )

Sect. 17

Fig. 17.5a, h. Dispersion relations of the optical modes of a zincblende crystal ZnS. (BALKANSKI,

1973)

steep branches in the (111 )-direction are due to the very strong and partially covalent Ti - 0 forces while several of the flatter dispersion curves are connected with the much weaker Sr - 0 forces_ In addition the polarizabilities of Sr


CD ® STRONTiUM

I (0,0,0) I I

I I

® • TITANIUM

: .' I .' -----@ (H~) @----,.-- -J/ @. I

II, : 0 OXYGEN

~ @-. (·Ho) , ?;:..-: _ ..... - .. ------.-- II (!-O.t) ,,- CD III (o,H) ... - ......

Fig. 17.6. Crystal structure of perovskites ABX 3

and ° have to be considered while all other parameters are small and only give a refinement of the curves.

The fact that the two different types of sublattice forces, namely Sr - ° and Ti - ° forces, are not completely balanced in the perovskite structure, is indicated by two different structural phase transitions which frequently occur in ABX3 crystals. The first one is related to the transverse mode at the R-point which becomes unstable against a slight rotation of the Ti06 cluster against the Sr-sublattice if T~ 105 C. This instability appears in all perovskites and it means that the transverse Sr - ° forces, or generally A - X forces are too weak to keep the BX6 cluster in a completely symmetric position. We note that the Sr-O distance in SrTi03 exceeds that in cubic SrO by 7 % which indicates that the equilibrium distance in SrTi0 3 is changed in favor of the strong Ti - ° forces. The second instability occurs only in oxydes and is due to a sottening of the 'Cochran mode' i.e. the lowest of the three infrared active modes at r with frequencies WI' wn and Wm (Fig. 17.7). In SrTi03 the phase transition can only be obtained by uniaxial pressure, while in BaTi03 and other crystals the phase transition occurs at convieniently accessible temperatures.

The three coupled TO-modes at r form a very interesting example for the application of an extended Kurosawa formula (15.77). Analysis as of this type has been given in terms of phenomenological coupling parameters by many authors; we refer the reader to the reviews by PERRY (1972) and BARKER (1970). Our interest is in the dynamical microscopic treatment of the infrared data as given by COWLEY and BRUCE (COWLEY, 1963, 1964, 1965; BRUCE and COWLEY, 1973; COWLEY and BRUCE, 1974; BRUCE, 1975). Since the soft r-mode tends to become instable against a displacement of the Ti-ions against the neighboring O-ions if the temperature is lowered, it seems to be obvious that an effective anharmonic Ti - ° potential should be able to describe the temperature dependence of w'ro' In order to obtain a consistent treatment of all anharmonic effects, including thermal expansion and infrared absorption and the phase transition of the R-mode, BRUCE and COWLEY (1973) choose anharmonic central Sr - ° and Ti - ° potentials up to the forth order. They obtain 6 parameters which they fit to 10 macroscopic entities. A particular gratifying result is obtained, then, for the reflectivity of SrTi0 3 (Fig. 17.8). Here, a 3 x 3


5

o

o 0 ()'5-( 0 0(-().5 0(-0-5 0-5-(0 05- ( 0

Reduced wave vector coordinate ( ~ )

Fig. 17.7. Lattice vibrations of SrTi03 (after STIRLING, 1972, 1976). W" w lI ' Will: TO modes at r (IR active)

90

26 30 30 SO 10 90 110 130 'oIMienqlh!l'm)

~~SO~30~~W~~~~--~~' 10 S F~qoJtnCy CTHz )

Fig. 17.8. Infrared reflectivity of SrTi03 Experimental results: Theoretical curve. (BRUCE and COWLEY, 1972)

Sect. 17 Infrared spectra of crystals with mixed ionic and covalent character

4

3

-;:: 2 ::I: I-...... 1

-"'--~ ~ , ~ 0~------------~:~\}--~--~--------------~~==9 ~ \ ! J{ -I : \./

-2 : . . ' .. " . I .~,

: -3 I, ,

-... -~/, .... -\ ,,' \\,,/ '-'

o 5 10 15 20 Frequency CTHz).

25 30 35

1.1.5

Fig. 17.9. Self-energy of the TO-modes in SrTi03 (BRUCE and CoWLEY, 1972). Solid line: imaginary part; dashed line: real part

matrix for the three coupled wTo-modes has to be solved, including the offdiagonal elements of the phonon self-energy (Fig. 17.9). It turns out, as first has been shown by BARKER and ROPFIELD (1964) in an experimental analysis, that these off-diagonal i.e. mode-coupling parameters are very important. The calculated line-widths of the three TO-modes seem to agree well with Raman measurements (FLEURY and WORLOCK, 1968). The frequencies of the three renormalized TO-modes at 300 K are indicated in the figure.

As in the simple diatomic crystal the cubic Ti - 0 anharmonicity dominates the infrared spectra. For the temperature dependence of the soft mode at r, however, the quartic Ti - 0 parameter turned out to be the decisive quantity quite in agreement with previous results of other authors (HOLLER, 1969; LOWNDES and RASTOGI, 1973). This quartic parameter turns out to be numerically very high so that the question raises (BRUCE and COWLEY, 1973) whether this quantity simulates a mechanism different from an anharmonic quartic Ti- 0 overlap potential. We shall come back to that point in our discussion of the Raman spectra of perovskites in Sect. 19. We note here that an alternative explanation of the ferro electricity in perovskites is based on a negative contribution of the electron-phonon coupling to the phonon self-energy instead of anharmonic phonon-phonon coupling (cf. KRISTOFFEL and KONSIN, 1973). The critical point in the analysis is the importance of the anisotropy of the forces in perovskites. While the macroscopic symmetry of the lattice as well as the local symmetry of the metal A and B ions is cubic, the local symmetry of the X anions is not cubic, since they see two neighboring B ions in a line but four A ions in a perpendicular plane. It has been known for a long time (SLATER, 1950), that, therefore, the local field at the X ion lattice site is strongly enhanced in the direction of the B ions. This structural Coulomb anisotropy provides the starting point for the necessary compensation of short-range repulsive, long-


range attractive and polarization forces (COCHRAN, 1960) in a soft mode. One might ask whether an inter-ionic anharmonic potential between B and X ions could be sufficient for triggering the phase transition in view of the fact that only the oxides seems to show the B - X type of phase transition. We shall therefore discuss the influence of an intra-ionic anisotropy in the oxygen polarizability in connection with the Raman spectra of perovskites. It seems that the covalency of transition metal-oxygen forces plays an important role in this case.

For further details on the structural phase transitions in perovskites we refer to recent reviews by SCOTT (1972), SHIRANE (1973), LINES and GLASS (1977), DORNER (1981), and FLEURY and LYONS (1981). Many data on the infrared properties of perovskites may be found collected in Landolt-Bomstein (9, 1975; 16, 1981).

d) Spectra in low-symmetry crystals. In this section we discuss the spectra of crystals with a macroscopic symmetry lower than cubic symmetry so that the dielectric constant 8 becomes a tensor. We may, then, divide the low-symmetry crystals into two classes.

The first is that type of crystals where the breaking of cubic symmetry leads only to a quantitative complication of the description without leading to any new qualitative aspect of the spectra. An example of this type are the crystals with wurtzite structure, such as ZnO and CdO, which have four particles in a cell instead of the two appearing in zincblende and therefore exhibit a doubling of the optical modes, a genuine anisotropy of the dielectric tensor and, correspondingly of the Lyddane-Sachs-Teller splitting of the optical modes which shows up in an anisotropy of the optical data, for example, of the reflection spectra. Since the difference between zincblende and wurtzite crystals is only due to the difference of second nearest-neighbor forces which are usually rather small we shall expect that the difference in the lattice dynamics and infrared spectra is also small so that, in a first approximation we may treat the wurtzite crystals as if they are still of zincblende symmetry (cubic approximation). The deviations from this picture may then be treated in a perturbational treatment. For example, we show in Fig. 17.10 the reflection spectra of CdS in the zincblende and in the wurtzite structure in order to demonstrate the small differences. We refer the reader for a discussion of the spectra in wurtzite crystals to SPITZER (1968) and BALKANSKI (1973).

A second class of crystals is obtained from those compounds where the derivation from the cubic symmetry is essential. Typical examples of this type are crystals with layered structures where the inter-layer forces strongly derivate from the intra-layer ones, or pseudo one-dimensional crystals such as selenium and tellurium which exhibit more or less strong covalent intra-chain forces with relatively weak inter-chain forces. We shall discuss this letter case a bit more in detail because of the general interest in strongly anisotropic crystals. The dynamical properties of selenium have recently been investicated by NAKAJAMA and ODAJIMA (1973), WENDEL, WEBER, and TEUCHERT (1975, 1976) and by HAMILTON et al. (1974), those of tellurium by PINE and DRESSELHAUS (1971) GIBBONS (1973) and E.R. COWLEY (1973). The trigonal crystals Se and Te may be viewed as intermediate crystals between molecular sulfur and metallic polonium (MARTIN and LUCOVSKY, 1974). Their helix-chain structure (Fig.


80.-------------------------------------------------.

v (cm-')

Fig. 17.10. Reflection spectra of CdS in wurtzite and in zincblende structure

r c

L Fig. 17.11. Chain structure of selenium and tellurium. (After WENDEL, 1975)

17.11) can be described as a distorted cubic structure with two intra -chain neighbors (distance f i) and four inter-chain neighbors (distance fa)' The ratio fa/fi

(S: 1.6, Se: 1.45, Te: 1.23, Po: 1) is a measure for the anisotropy of the forces which can be suppressed with sufficiently high pressure where Se (140 Kbar) and Te (40 Kbar) undergo an insulator-metal phase transition. The anisotropy also is reflected in the elastic constants (STUKE, 1969; FJELDL Y and RICHTER, 1974) the dielectric constants and the effective charges (WENDEL, 1975). A fit with a 14-parameter shell model with valence forces is still not quite satisfactory for the quanstitative description of the effective charges, the dielectric constants


50,-----------------------------------------,

E II c

;;.!!

c: 10 ., en a '0 E 0 c;; >

'" c: 0 40 x 'x

E 1. c ., ~

30 a::

20 .. v", __ ~....,._a

10 b

050

Wellenzahl (em-i)

Fig. 17.12a, b. Reflection spectra of trigonal selenium (GEICK et a!., 1970). a) Reflectivity for nearly normal incidence versus wavenumber, experimental data (solid curve) and calculated by means of a classical dispersion formula ( x), (E II c) b) Reflectivity for nearly normal incidence versus wavenumber, experimental data (solid curve) and calculated by means of a classical dispersion formula ( x),

(E1.c)

and the LO - TO splitting, probably due to a certain delocalization of the electronic charge between the ions.

In Fig. 17.12 the reflection spectra of selenium are shown as investigated by GEICK et al. (1970). One infrared active mode for Ellc[wTo(I;-)~104cm-1] and a second for E1-C[WTO(I;1)~140 cm- 1] are clearly seen in the spectra while the third mode [WTO(I;2) ~ 233 cm -1] has too small an oscillator strength to be found in the reflection spectra. There is a fourth mode at r which is Raman active only [wTo(r;,)~236 cm- I ]. The frequencies of the r-modes determined from a classical oscillator fit or transmission minima are in very good agreement with the neutron inelastic scattering data. There is no indication that the difference between the infrared eigenfrequencies and those found in inelastic neutron scattering at r is strong enough to be measured.

The distinction between covalent, ionic and polarization forces, their anisotropy and their trends in a family of low-symmetry crystals such as the group VI elements with increasing metallicity as a function of the main quantum number and a corresponding behavior of effective charge tensor etc. is known for a large variety of crystals and implies similar problems with respect to a proper treatment of lattice vibrations and infrared absorption as have been discussed for Se and Te.


e) Spectra of amorphous semiconductors. Amorphous systems are usually described in terms of continuous random-network models (insulators) or randomclose-packing phases (metals). An alternative possibility is to define an amorphous solid phase by optimization of short- and medium-range order ('glasses') while longrange order is lacking. The case of tetrahedrally co-ordinated systems (a-Si, SiOz, etc.) has been discussed in detail by many authors [for recent reviews refer to WONDRATSCHEK (1965, structural properties of glasses); BRODSKY (1979, electronic and vibrational properties); W.A. PHILLIPS (1981, low temperature properties)]. Dynamical and topological arguments are often used to show that very large regular molecules can be constructed which exhibit many of the properties of glasses but do not possess translational invariance. The problem of the excess specific heat at very low temperatures and the role of two-level systems is of particular interest for the understanding of the microscopic processes in amorphous systems (see HUNKLINGER, 1981).

With respect to vibrational spectra, the main difference of amorphous systems as compared to crystals is the lack of selection rules due to the breakdown of translational invariance and point symmetry. One expects one-, two-, and many-phonon spectra more or less resembling the corresponding density-of-states spectra.

Vi 'c :J

5r---------------------------------~

-e c Q)

~ I

U .... o til

4

3

16 2 ti '0 :c'iii c

L X,E W

t t t

Q) 0 °300~~==============~==========~

E 240 1 a-Ge(1) pure u ....... 2 a-Ge(2) H: 6.1 0\ % == C 3 a-Ge(3) H: 7.40\ % Q)

180 'u .... .... (l)

0 u c 120 0

0.. ::; til 60 .0

<!

o Wave number (l/cm)

Fig. 17.13. Vibrational absorption spectra of pure and hydrogenated a-Ge. Density of states: WEBER (1977)


In their review on the vibrational properties of amorphous semiconductors LUCOVSKY and HAYES (1979) distinguish between different categories of models: (I) models based on comparisons between the vibrational spectra of crystalline and amorphous solids of the same composition (Si and IX-Si, for example); (II) calculations based on very large clusters containing hundreds of atoms; (III) 'exact' calculations based on small clusters (e.g. Bethe-Iattice method); and (IV) molecular models with decoupled molecular clusters. It seems that each of these different models is able to explain some specific aspects of the dynamics of amorphous systems while having difficulties with other properties.

To be specific, the infrared spectra of IX-Ge and IX-Si are shown in Figs. 17.13 and 17.14, respectively, together with the one-phonon densities of the perfect crystals. Surprisingly, the critical points of the one-phonon densities show a nearly one-to-one correspondance to the maxima and kinks of the experimental spectra, in particular if a weak 'scaling' of the crystal's frequencies has been assumed. These findings in principle support the 'local' models of type (III) and indicate that the main structure in the frequency spectrum originates with a rather small subset of 'molecular' frequencies. - A second point to note is the existence of dipolar effective charges for all infrared

":§ 'c ::J .d

.2 Vi

I u .... 0 III

'" '0 1ii '0 Z' 'iii c: '" 0

E u ......

~

c:

'" 'u .... .... Q) 0 u c:

~ a. 0 III .0 <{

5

4

L 3 t 2

0 300

240

180

120

60

o

X,I: W

• •

1 a-Si 0) pure 2 a-Si (2) H : 19.6 at % 3 a-Si (3) H:24.2at% 4 a-Si(4)H:15.0at%

Wave number (l/cm)

Fig. 17.14. Vibrational absorption spectra of pure and hydrogenated a-Si. Density of states: WEBER (1977)

Sect. 18 Raman scattering from ionic crystals 229

frequencies in the amorphous semiconductors in contrast to the perfect crystals which do not exhibit anyone-phonon IR spectra (and one Ramanline, only). Looking for a microscopic argument to explain this observation one should notice that the 6-rings of zig-zag ('seat') type which build the diamond lattice have no resulting dipole moment per ring while the other possible, 'boat'-type 6-ring configuration (which one knows from the c-direction in zincblende lattices) exhibits a non-vanishing net dipole moment. In a simple cluster model, with these 6-rings and 5-rings only (DANDOLOFF et aI., 1980a), a natural explanation of the above observation is obtained. A combination of this dynamical model with some general stability arguments first discussed by J.e. PHILLIPS (1979) seems to provide a promising starting point for a microscopic dynamical description of amorphous or, more precise, glasseous systems. (DANDOLOFF et al. 1980b, J.e. PHILLIPS, 1982).

For further information on infrared (and Raman) spectra of amorphous systems the reader is referred to the literature mentioned above and to the reviews by WEAIRE (1980) and by WEAIRE and TAYLOR (1980).

18. Raman scattering from ionic crystals (cf. WOLKENSTEIN, 1973; CARDONA, 1975). The infrared spectra of ionic crystals are mainly determined by first order resonant absorption from dispersion oscillators with their anharmonic line widths leading to multiple phonon structure in the tails of the absorption. On the other hand, in the strongly covalent crystals of high symmetry such as diamond, etc., genuine second or higher order non-resonant spectra may be observed. The first order resonance, in this case, is situated far above in the optical frequency regime of the energy band transitions. In the model description of these spectra this means that the role of the effective charge of the non-linear dipole moments is played by an electronic charge with a mass about 104 times smaller than the ionic masses are. The infrared spectra of the covalent crystals are low-frequency off-resonant spectra and exhibit simple relations to combined densities, critical points, etc., which have been discussed in Sect. 16.

This situation is to a certain extent reminiscent to Raman scattering by ionic crystals. In particular the Raman spectra of cubic ionic crystals with inversion symmetry do not show a first order Raman line as a result of inversion symmetry. Their second order spectra are all far below resonance, i.e. the band gap transition in the UV. Similarities with infrared spectra are very superficial and the analysis of infrared spectra does not tell us much about Raman scattering in the same crystal. The Raman spectra are, in addition, of greater complexity than the infrared spectra since those have at least three different parts with symmetries ~+, ~1, and I;~. While ~+ shows the full symmetry of the crystal, the other two components have only quadrupolar symmetry. A quantitative explanation of the spectra should a priori be more difficult than one of infrared spectra. The general features which are still similar in both types of spectra are, first, the cutoffs of the two-phonon bands which are often clearly seen in both cases, and, secondly, a rather pronounced structure which can be associated with critical points of the two-phonon joined density and is in the Raman spectra governed by overtone contributions and not by combination spectra which determine infrared absorption.


If the inversion symmetry is missing, the ionic crystals show first order lines, but the corresponding 'Raman' oscillators don't play the role of resonant dispersion oscillators. The reason is that the Raman phonon spectra only appear as sidebands of electronic transitions and not via direct scattering of the light by an ionic Raman oscillator. This (infrared) ionic Raman effect is very weak due to the 0)4 factor in front of the cross section and has not been observed so far. The possibility of its observation under favorite circumstances has recently been discussed by T.P. MARTIN (1974). Its realization would be of enormous interest for the theory since in this case the anharmonic phonon-phonon coupling would playa similar but complementary role as in infrared absorption.

On the other hand, the resonance Raman effect in the visible regime of the optical spectrum has been investigated very intensively during the last decade (refer to CARDONA, 1975; RICHTER, 1980). Its analysis provides interesting details of the electron-phonon coupling connected with particular electronic band-band transitions and allows a determination of the coupling parameters for these specific cases. It provides therefore an important information in addition to the off-resonant spectra. These latter ones are in the center of the present investigation where the vibrational aspect of Raman scattering is emphasized. The resonance Raman effect is briefly discussed in Sect. 9 g.

a) Raman spectra of cubic ionic crystals. In Figs. 18.1 and 18.2 the second order Raman spectra of two specific families of simple ionic crystals are shown. Figure 18.1 contains the spectra of three different alkali chlorides. The zero

-III -'2 ::J .d .... .Q

cm-1

cm-1 Fig. 18.1. Second order Raman spectra (I;+ +4I;.i +r;~) of alkali chlorides. (After KRAUZMAN 1970 and RIEDER, 1974). The spectra are shiftet one against another by 55 cm- 1 (KCl and NaCl)

and by 58 em - 1 (RbCl and KCl)

Sect. 18

o

Raman scattering from ionic crystals

8 <I: I-

N

MgO

, \,

Coo \ \

') I I I I I I I I I I I I

:J8 c -0-01-0 1-+1-N~N

...... ~./ /1

..... $.:::-_1

SrO

BaO

'" e:: ';;; '" 8 e 8 2 u

0 ." g g .1- e::

N .& N N

w

;:

g N

100)

Fig. 18.2. Second order Raman spectra of earth alkaline oxides, (From RIEDER et ai., 1973)

231


points of the frequency scale are shifted against one another in order to demonstrate the similarities in the general features of the spectra. Corresponding arrangements of spectra can be made for fluorides, bromides and iodides. In all cases the second-order spectra show a strong decrease above the high-frequency limit of the 2 TO regime and exhibit a very weak residual scattering in the following 2 LO frequency regime. This behavior parallels the rather low twophonon density of states in this latter regime.

The second order spectra of the alkaline earth oxides, as shown in Fig. 18.2 and arranged in a manner similar to those in Fig. 18.1, behave quite differently. In addition to the two-phonon spectra of transverse phonons they exhibit a very strong scattering in the 2 LO regime. This property seems to be very typical for all oxydic crystals and can even be used to distinguish the oxydic perovskites from those which do not contain oxygen (refer to Sect. 18 c). A similar behavior has been observed in the case of hydrides (LiH: CUNNINGHAM et aI., 1975; LiD: LAPLACE, 1975).

The similarities in the spectra suggest a specific mechanism for the change of the polarizability in these crystals. In the case of alkali halides we shall see that inter-ionic nonlinear polarizabilities are governing the spectra. It seems that this fact is in agreement with a simple inspection of the band structure. While in the case of linear polarizabilities the states below and above the ionization energy are of equal importance (DALGARNO, 1960) the Raman polarizabilities for second order scattering contain two more energy denominators which favour the relative weight of the low energy transitions. We may therefore expect that, in a first approximation, transitions between the (anion p6) valence band and the first (cation 82) conduction band should play an important role. In a formal description, these charge transfer transitions correspond to inter-ionic nearestneighbor polarizabilities which, actually, describe the Raman spectra of alkali halides quite well (refer to Sect. 18b).

One might argue that the earth alkaline oxides should exhibit a similar behavior since their electronic band structure is similar to that of the alkali halides. In particular, the valence and conduction bands originate also from anion p and cation 8 states, respectively, and the energy gaps show comparable or often higher values (:<: 10 eV) since the Coulomb splitting for a divalent ionic crystal is higher than for a monovalent one. Surprisingly, the oxides display a strong intra-ionic scattering which corresponds to completely localized transitions at the oxygen sites involving very weak charge transfer processes only. This means that a simple band structure analysis is un sufficient for the understanding of the spectra.

We may explain this unexpected behavior considering the results of Hartree-Fock calculations for the band structure of MgO (refer to LIN et aI., 1974). Here it turns out that the insulating gap is reduced by a factor of two due to strong correlation effects. In terms of an effective one-electron description we may express this by modifying strongly the wave functions of the conduction band states in the direction of great admixtures of oxygen states which generate intra-ionic transition matrix elements. The same effect was described in Sect. 14 where it was related to the non-rigidity of the 0 - - 2 p shell. This allows for an adiabatic re-adjustment of the 2p wave functions in response to a displacement


r

Fig. 18.3. Radial part of the 2 p wave function of 0 - - as a function of the lattice constant (schematical)

of neighboring ions (Fig. 18.3). The shells of monovalent ions, such as CI-, Be, are too rigid to show this effect. We shall therefore expect that in all those crystals where ionic constituents display a radius-dependent polarizability strong intra-ionic non-linear polarizabilities become important. In addition to H - and 0 2 - a very interesting cancidate seems to be N 3 -, which may exist in a few particular crystals such as Li3N (refer to BRAUER, 1938; CHANDRASEKHAR

et al., 1976; RABENAU, 1978; KRESS et al., 1980). Unfortunately, at the time being, no calculations, exist which explain intra-ionic polarizabilities in ionic crystals explicite1y in terms of microscopic matrix elements. Therefore, the foregoing discussion may be considered as a plausible argument only. We are now going to discuss the different results obtained for individual crystals. First, we recall the results of the symmetry analysis for octahedral crystals given in Sect. 13 (refer to Table 40.2). The polarizability tensor P contains three symmetric components r1+(A 1g), ~~(Eg), and I;~(F2g) and one anti-symmetric component rl~(Flg) which usually is neglected in off-resonant scattering due to the factor (OJjQ)2 in front of P since the phonon frequency OJ is small as compared with the frequency Q of the external light. The matrices P{r) are built up from diagonal and off-diagonal elements a, band c, d, respectively, which finally determine the components of the Raman scattering tensor i (refer to Table 40.4).

Since in the crystals under consideration every ion is a center of inversion symmetry all lattice modes at r are Raman inactive. Weare therefore concerned with the second order spectra (third order spectra are generally too weak to be observed). As in the case of infrared spectra, one might try to relate the main features of the spectra to specific critical points of the joined density of states. The selection rules for the main overtones and combinations for Raman processes in rocksalt crystals are given in Table 40.6. We recall that in alkali halides and other cubic diatomic crystals the main peaks and shoulders of infrared spectra can be interpreted in terms of combinations of phonons at the Lpoint. The reason has been found in a playing together of the selection rules with the saddle point character of the phonon dispersion and the q-dependence of the matrix elements determined by a central two-ion potential (refer to Sect. 15. d). A similarly simple and general analysis of the Raman spectra while very desirable seems not to exist at the time being although very often overtones and combinations of phonons at the X- and the L-point have frequencies close to experimental maxima or kinks in the Raman spectra (refer to KRAUZMAN,


1970). This does not mean that the main contributions to the scattering intensities actually originate from two phonon processes in the neighborhood of the X- or the L-point. It rather indicates that there exist unknown correlations between the frequencies of these particular two-phonon processes and"the still hidden dominating ones somewhere in the Brillouin zone which are governed by some Raman 'potential'. The knowledge of this potential would be equivalent to an explicit description of the non-linear electron-phonon interaction in ordinary space.

An interesting group of crystals are the hydrides, oxides and perhaps nitrides where the intra-ionic polarizabilities of the anions dominate the spectra. Since the existence of the corresponding strongly localized electron-phonon coupling is always indicated by the intense scattering in the 2 LO phonons regime and in favorite cases, such as MgO, a quantitative description of the spectra with only one adustable parameter can be achieved (refer to Fig. 18.6) a detailed analysis of the origin of the different parts of the spectra in terms of specific two-phonon processes becomes possible (see below). Furthermore the dominance of the volume scattering leads to the conclusion that the genuine quartic part of the local electron-ion potential (Diagram 14.1d) dominates over the iterated cubic process (Diagram 14.1c). We note that this statement refers to the displaced ionic configuration and might not without caution be compared with results obtained for the equilibrium configuration.

It is interesting that an analysis of this type is still successful in the case of tetrahedral semiconductors such as the zinc chalcogenides (see Sect. 19b). The reason may be found again in the flexibility of the ground state wave functions which in these crystals are hybridized to some extent with wave functions of neighboring ions. Due to this open-shell behavior, it is impossible to determine a well-defined linear ionic polarizability. The change of the crystal's polarizability is than, as in the case of oxygen, rather well localized at the ion lattice sites and appears as an "intra-ionic" mechanism when described in a model theory.

IX) Raman spectra of alkali halides. After the early measurements of the unpolarized spectra of NaCI by KRISHNAN (1945) and its interpretation by BORN and BRADBURN (1947) in terms of an rigid ion model with weighted twophonon densities there were few investigations mainly using mercury arc lamps (e.g. WELSH et aI., 1949, for NaCI, with a deformation dipole model analysis by KARO and HARDY, 1966) up to the advent of lasers about 15 years ago. Since then, a still increasing amount of Raman spectra of ionic and other crystals has been recorded. Here, we focus attention on rather new data, in particular such which have been analysed in terms of formal or model parameters.

An investigation of numerous diatomic ionic crystals has been presented by KRAUZMAN (1969, 1973). Among them the spectra of KBr attracted particular attention. The main reason is that the lattice dynamics of KBr is wellknown both experimentally and theoretically (COCHRAN et aI., 1960; COWLEY et aI., 1963) so that the idea of a quantitative analysis of the spectra seems to be tempting. Of particular interest is the fact that in KBr Gust as in NaI) the longitudinal optic branch near the L-point exhibits a strong intra-ionic 'breathing' deformability (SCHRODER, 1966; NDsSLEIN and SCHRODER, 1967). One


might therefore speculate about a non-linear extension of the breathing shell model (refer to Sect. 4) in terms of non-linear intra-ionic polarizabilities of breathing (I~+) and dipolar (~5) type since both seem to po cess a sound basis in the harmonic lattice dynamics of alkali halides. This idea has been carried out in detail by BRUCE and COWLEY (BRUCE and COWLEY, 1972, 1973; BRUCE, 1973) with emphasis on the spectra of KBr and NaC!. We obtain their results by specifying the formula of Sect. 14, (14.17), for rocksalt crystals and by adding the contribution of a breathing deformability to the Raman tensor:

l!.~)(q,jti2)= CI {a(K) [bafl I wP)w/2) " Y

+wa(l) wfl(2) +Wfl(l) wa(2)] + 3 b(K) bafl wa(l) wfl(2)

+ C(K) v(l)( cJiss 1 (K, q))afl' v(2) + d(K) bafl v(l) v(2), (18.1) with

Here C denotes a normalization constant, a(K) and b(K) are the fourth-order coupling parameters CXlcJissss(K) with spherical and cubic symmetry, respectively. The second term originates from the cubic environment of the ion and is zero for a free ion. d(K) = cJiSBBS(K) denotes an analogous forth-order term with respect to the breathing deformability. An additional term with a coupling parameter of the form cJiSBS cJii.;/ cJiSBS describing contributions from a cubic breathing' potential is given by the third term of (18.1) with a coupling constant c and a q-dependent propagator cJiis 1 which makes it different from the fourth term.

In their discussion of NaCI and KBr Bruce and Cowley neglected contributions from Na + and K + which may be treated as rigid ions in the harmonic shell model. In addition, they found that the cubic part ocb(2) was not very helpful in improving the ratio of intensities for different symmetries so that they finally used only the two parameters a(2) and e(2):

~~)(q,j,j') = C {a(2) [w(2) w(2') + wxC2) wx(2')] + [e (2) ( cJiss 1 ):x + d(2)] v(2) v(2')}

~~)(q,j,j') = C· {e(2)( cJiss 1 ):y v(2) v(2') + a(2) [wx (2) wy(2') + wy(2) wxC2')]} (18.2)

in a self-explaining short notation. As inspection of (13.12)ff. shows, the important 'trace' terms in w2 and V 2

contribute in (18.2) quadratically to the volume scattering tensor i(~+), linearly to i(~1) and nothing to i(I;1) which leads to a parallel sequence of intensity ratios and to a dominance of the volume scattering. This is, of course, a consequence of the highly isotropic intra-ionic scattering mechanism. A second point to note is that the trace terms favour the scattering by longitudinal phonons since these phonons are strongly influenced by the breathing and dipolar deformabilities and vice versa lead on their side to the strongest shell displacements W(2) of the polarizable ions irrespective of the fact that the Raman tensor P is connected with a transverse pohirization of the electrons. If the polarizable ion is heavier than the non-polarizable one as it is the case in NaCI and in KBr the dipolar scattering mechanism is most effective in the longitudinal acoustic branch (LA)

236

a

30 100


, I I I I

~ /2LO(U\ .k --~ I \1

200 cm-! 300

c

I I I I

b

30 100 200 em'!

Fig. 18.4a-c. Second order Raman spectra of KBr. (From KRAUZMAN, 1972)

Sect. 18

300

while the breathing term is dominant in the LO regime. We expect therefore correspondingly strong contributions in the overtone regime 2 LA and 2 LO, respectively.

In Fig. 18.4 the experimental results for KBr obtained by KRAUZMAN (1973) are compared with a calculation by BRUCE and COWLEY (1972). These authors have fitted the three parameters a, c and d of (18.1) (ex: to L 1, L 3 , L 4 , in their notation) to the experimental data. Since the absolute intensities are unknown, only the two ratios cia and dla have to be determined. In view of this small number of parameters and the experimental uncertainties the results are not unsatisfactory. All important features are reproduced within an intensity factor ;52. Some of the calculated peaks in the upper part of the ~+(Alg) spectrum do not seem to have an experimental counterpart. A better result was obtained by BRUCE (1973) for NaCI with only the two quartic parameters a and d. Again, agreement seems to be worse in the high-frequency part of the ~+ spectrum. -The results of BRUCE and COWLEY for KBr have been disputed by KRAUZMAN (1973). In his discussion he used the same breathing shell model but calculated the Raman tensor in a more formal way by considering first all types of expansion coefficients of second order into ionic (u), shell (w) and breathing (v) displacements which lead to terms in u u, u w, u v, etc. He found that even neglecting the breathing mode of the K + ion one obtains in a nearest-neighbor approximation 23 parameters contributing to the ~+ and the ~i spectra while 13 others contribute to the I;~ spectrum. There is, actually, a further restriction


due to the adiabatic condition but a fit in this parameter space certainly would not give a meaningful result. Krauzman restricts then the analysis to the 5 constants which contribute to the ~+ and ~i spectra and involve only core displacements u. Since the ~i which depends only on differences of those parameters is rather weak as compared to the ~+ spectrum he neglects the ~i part by assuming all 5 parameters to be equal. He then obtains the curve shown in Fig. 18.4 which compares favourably to that of BRUCE and COWLEY. Furthermore, as for the I;~ spectrum, he shows that even a single term in ua(K +) up(Br -) would give a nice fit to the experimental spectra. Finally KRAUZMAN discusses a breathing term equivalent to that ocd(Br-), (18.1), which gives a rather isolated peak at the position of the highest peak (near 2 LO(L)) of the theoretical ~+ spectra by BRUCE and COWLEY (Fig. 18.4). This contribution to the spectrum is, however, strongly modified in Bruce and Cowley's calculation by the third term in (18.1) occ(Br-) which has actually an opposite sign as compared to d(Br-) and gives a strongly frequency dependent correction to the spectrum due to the propagator cJ>Ss 1. KRAUZMAN concludes from his discussion that there exists no evidence for the non-linear extension of the breathing and dipolar deform abilities in the Raman spectra of KBr. This argument is not as strong as it looks at the first glance. In addition to what has been stated above on the frequency-dependent breathing polarizability it should be noted that the electronic degrees of freedom wand v may be expressed in terms of core displacements u of the same ion, the neighboring ones, etc. This means that nonlinear intra-ionic polarizabilities in w2 and v2 can be completely replaced by formal inter-ionic polarizabilities in u2 including nearest, second nearest, etc. neighbors. In favorite circumstances a nearest-neighbor approximation may give quite a satisfactory fit to the experimental data.

We have performed this very detailed discussion in order to demonstrate the difficulty to obtain reliable conclusions from Raman spectra about the coupling parameter based on one ore two specific crystals only. It seems therefore to be necessary to look for trends i.e. some systematic behaviour in crystal families with similar physical properties. In the case of alkali halides a systematic study of the known second order Raman spectra of alkali halides has been carried out by HABERKORN et al. (1974, unpublished). This investigation gives strong support to Krauzman's conjecture for all alkali halides. With the only exception of NaI where intra-ionic polarizabilities may contribute to the spectra by about 20 % it seems that inter-ionic polarizabilities dominate the spectra. A very strong argument in this direction stems from the fact that in alkali halides with lighter anion masses no strong Raman scattering in the 2 LO regime can be observed. A striking example is RbCl (Fig. 18.1). It is hard to understand that in this case the intra-ionic dipolar Raman polarizability is very weak (according to the scattering above 2 TO(r)) while in NaCI the corresponding part in the 2 LA regime is claimed to originate just from this mechanism.

In the introduction of this section a microscopic argument was given by stating that the change of the polarizability of rigid closed-shell ions, such as the anions in alkali halides are, needs a charge-transfer i.e. an inter-ionic mechanism. It will be seen in forthcoming sections that the intra-ionic mechanism can be well established in crystals with open-shell ions. An impressive example is


given by the earth alkaline oxides discussed in the next section. Beforehand, a few more calculations of alkali halide spectra shall be reviewed.

First, we note investigations of these spectra in terms of critical points in the two-phonon density of states using the proper selection rules for Raman scattering (Table 40.7 and 40.8). Many of them have been carried out by KARO and HARDY (1966a, b, 1969) and by KRAUZMAN (1969, 1970, 1973). For other investigations of this type refer to the article by BIRMAN (Vol. XXV /2 b of this Encyclopedia). In contrast to the case of infrared absorption where the main features of the spectra seem to be related to critical points of combination branches at the L-point, the situation is more complicated in the case of Raman scattering where many different critical points seem to be important for a discussion of the spectra. Probably, this is due to the lack of a dynamical model for the electron-phonon interaction in these crystals. As a consequence, a critical point analysis may often mean an artificially correct 'labelling' without any relation to the origin of the scattering intensity in that frequency regime.

More important are investigations which try to evaluate the formal expansion parameters ~py~ by fitting them to the experimental data following the first paper by BORN and BRADBURN (1947). Here, the papers by HARDY, JASWAL and their co-workers are useful, in particular a paper by CUNNINGHAM et a1. (1974) which discusses in detail the second-order spectra in alkali fluoride crystals. It contains also a critical review of proceeding papers and it reestablishes the result of KRAUZMAN for KBr that in a Born-Bradburn treatment the nearest-neighbor polarizabilities ~p(12) are more or less sufficient to describe the three components of the Raman spectra. This means 5 parameters for the 11+ and the 11i spectra and 3 more parameters for the I;,~ spectra. In this latter case, 2 second nearest neighbour (n.n.) parameters seem to give a substantial improvement of the description. - An analogous analysis for cesium halides, i.e. alkali halides with cesium chloride structure has been given by AGRAWAL, KIRBY and HARDY (1975). Here we have (as compared to the NaCI structure) 10 (instead of 8) n.n. and 8 (instead of 15) second n.n. polarizability coefficients for each particle in a cell. AGRAWAL et a1. assumed that the polarizability is only affected by the 'bond' length of nearest and 2-nd n.n.-ions. This reduces the number of 2-nd n.n. parameters from 16 to 6. This assumption, although plausible at the first glance, seems to be very debatable. The 16-parameter fit gives, not surprisingly, a good description but demonstrates drastically the lack of a physical understanding of the electron-phonon coupling in the alkali halides.

A step into the direction of a derivation of the Raman spectra from some general features of the shell model has been taken by PASTERNAK, COHEN and GILAT (1974, 1975). They follow Cowley's original (1964) representation of the anharmonic potential energy with coefficients «fJ(u, u, w, w), and reduce the number of parameters by assuming that only nearest and second nearest neighbors are involved, that only radial changes of the polarizability occur, i.e. parallel to those displacements which are parallel to the interconnecting line between the ions, and that the second nearest neighbor contributions are isotropic i.e. independent of the cell index 1. This leaves them with four


independent parameters, one of which may be fitted to the intensity of the ~+ spectrum. While the representation of the NaCI and KBr spectra may be considered to be satisfactory, the calculation of the MgO spectra is missing the high frequency part. Unfortunately, the treatment of the polarizability in a mixed representation (Born-Bradburn parameters for shell model phonons) makes it very difficult to interpret the parameters in physical terms. In the case of MgO, the analysis seems to be wrong because the one-parameter intra-ionic analysis of the spectra by HABERKORN et al. (1973) gives a better agreement with the observed spectra (refer to I8.allI).

We remark that in all spectra discussed so far (except for MgO) the intensity in the 2 LO regime is very weak. This intensity is related to intra-ionic and second and higher order neighbor polarizability coefficients.

o 100 200 300 1,00 500

wave number shiff (em-I) --~

Fig. I8.Sa. Stokes Raman spectra of Agel at 77 K for 5145 A excitation. Spectra (a), (b), and (c) were measured in z(x, x) y, z(x, z) y, and z(y, x) y configurations, respectively. Spectrum (d) is the r; spectrum computed from (a)-t(b)

-(c).

100 o 100 200 300

wave number shiff (cm- I)--------<-

Fig. I8.5b. Stokes and anti-Stokes Raman spectra of AgBr at 77 K for 4880 A excitation.

(See Fig. 15.a caption for details.)


f3) Silver halides. First, we discuss the spectra of silver halides. The unpolarized spectra of AgCl and AgBr have been measured by BOTTGER and DAMSGARD (1972) while the polarized spectra have been analyzed by VAN DER OSTEN (1974) (Fig. 18.5). While the lower part of the unpolarized spectrum shows a close connection with the two-phonon density of states (FISCHER et aI., 1972) the high-frequency part in the 2 LO regime is much stronger than one would expect from the density of states. While this particular feature is reminiscent to the situation found in the alkaline earth oxides (Fig. 18.2), a striking difference occurs in the intensities since here the quadrupolar scattering (r1i and r2~) turns out to be stronger than the volume scattering (r1+). Probably, this is due to the quadrupolar deform ability of the Ag+ ion (r1i) which leads to an unusually strong first-order impurity induced Raman spectra with r1i symmetry of Ag+ impurities in alkali halides (MULLER and KAISER, 1972) and may be responsible for the enhancement of quadrupolar scattering in the second-order spectra, too. The 2 LO spectra could, nevertheless, originate from the strong anion-anion overlap in those crystals.

/,) Alkali hydrides and alkaline earth oxides. The Raman spectra of the alkaline earth oxides (MgO, etc.) and those of LiH and LiD are different from those of the alkali halides by showing a strong scattering intensity in the 2 LO frequency regime (Fig. 18.2). We have mentioned above that the anions 0-and H - differ from halogen ions, such as F -, Cl-, etc., due to the change of their polarizability with the ionic radius. This instability of the free ions results

- Experiment

MgO --- Theory

Wavenumber Ccm-t) Fig. 18.6. Second order Raman spectra of MgO. (From BUCHANAN et a!., 1973)


in a strong intra-ionic non-linear polarizability. We are now going to exemplify this statement.

A one-parameter calculation of the three spectra of MgO has been given by HABERKORN, BUCHANAN and BILZ (HABERKORN, BUCHANAN and BILZ, 1973; BUCHANAN, HABERKORN and BILZ, 1974) (Fig. 18.6). Only the spherical part of the intraionic quartic polarizability, a(2) in (18.6) has been used. The result is surprisingly good in view of the complexity of the shape of the spectra. No interionic (nearest- or second nearest-neighbor) shell model coupling parameters are assumed in contrast to the approach by PASTERNAK et aI. (1975). Similar results are obtained for CaO (BUCHANAN et aI., 1974) and SrO (RIEDER et aI., 1975; MIGONI et aI., 1975). In the latter case the importance of inter-ionic polarizabil-

VI C

~ C

c

'" E

'" c::

Lithium hydride isotopes, Raman spectra 10K

o Wave number (em-I)

Fig. 18.7. Unpolarized second-order Raman spectra of lithium hydride measured at 10 K for the three isotopic compounds 6LiH, 7LiH and LiD. The observed frequency ratios for the 3 isotopes fall into 3 distinct groups: peaks at high frequencies correspond to predominantly proton or

deuteron motion (ex:YMH or YMo) those at low frequencies to Li+ motion (ex:YM(Li) or in-phase

motion of both Li+ and H- ions (ex:YMH+MLi); and those at intermediate frequencies to lithium and hydrogen anti-phase motions (ex: to square root of reduced mass jl=MH-MLi/MLi +MH)' The simplicity of this categorization implies that two-phonon transitions involving different types of modes to not contribute, and that L point zone-edge modes (with one or the other type of ion being

stationary) are mostly responsible for observed spectra (ANDERSON and LUTY, 1982)


ities in addition to the intra-ionic terms has been demonstrated. The numerical values of the fitted parameters indicate, that a long-range Coulomb-type of interaction may be important. - A similar situation seems to exist in LiH and LiD. The spectra have been measured and analysed in terms of formal parameters by JASWAL et al. (1974), by LAPLACE (1973, 1975) and by ANDERSON and LUTY (1982) (see Figs. 18.7 and 18.8). - HABERKORN et al. (1974) were able to explain the high frequency part of the spectrum again with the intra-ionic spherical quartic polarizability. The low-frequency part indicates contributions from inter-ionic terms. It seems that the idea of intra-ionic polarizabilities is quite well established in the above-mentioned cases.

JASWAL (1975) has pointed to the question of the inter-relation of formal (,Born-Bradburn') and shell-model ('Cowley') parameters. Of particular interest is this question in cases like MgO, where a single shell-model parameter is able to give a satisfactory description of the spectra.

The interrelation of formal and shell-model parameters may be described by using the adiabatic condition which relates the relative shell displacements w of a polarizable ion to the displacements of the lattice ions, u. From (4.18) follows

(18.3) K',K"

A sufficient approximation for our purpose should be a nearest-neighbor potential which leads to (refer to Sect. 4)

T~R and S=R+K (18.4) and

(18.5)

Ro=A+2B, RAq)=Acosqxa+B(cosqya+cosqza), (18.6)

etc. With K2 ~ Ro, this leads to

w",(2Iqj) R :oK (U",(llqj) cosqx a + U",(2Iqj)). ° 2

(18.7)

Here, we have 'assumed that only the anion is polarizable (K 1 ~ (0). Introducing (18.7) into (18.2) gives the following 'formal' parameters, (refer to (14.17)),

P",p(q,jj')= L P",pyo(lIKK') [ -m;ley(Klqj)eo(KIj'-q) 1r(1(', yo

R2 P"x,xx(41 11)=3a(2) ( ~ f P"x,xx(OI22)

Ro+ 2

R2 P"x,xx(OI12) =Pxx,xAOI21) = - 6a(2) (Ro + ~2)2 •

(18.9)

(18.10)

The phase factors in (18.8) are, for (1IKK), (4111): cos2 qxa, (0122):1, (0112) =(0121): cosqxa, respectively.

Sect. 18

U1 C

'" C

c d

E d

IX:

Raman scattering from ionic crystals 243

Lithium hydride isotopes,Raman spectra at 300 K

Wave number (cm-1)

Fig. 18.8. Same as Fig. 18.7 at 300 K. The corresponding frequencies and isotope shifts are similar as in Fig. 1. The lowest frequency line, which disappears and is absent in Fig. 1, is a difference

mode

Equations (18.9)-(18.10) show that the intra-ionic quartic polarizability of an anion (K = 2) leads automatically to formal expansion parameters in the cation and anion sublattices and to interionic (12) parameters. Even, if the harmonic forces are restricted to nearest-neighbors (A, B), the range of these forces in the cation sublattice is spread out to 4-th nearest neighbors (2 cos2qxa=2 - cos 2qxa) while the forces in the anion sublattice are at least including 2-nd nearest neighbor forces since otherwise the condition of translational invariance cannot be fulfilled:

I. K'

= - pxx.xx (0121)- Pxx,xx(1121)- ... , (18.l1)

or 3 a(2) = 6 a(2) - ... (no solution for a (2)=1= 0).


This calculation shows why authors in analyzing spectra of hybrides and oxides have to include rather far-reaching polarizability coefficients without being able to interpret the physical nature of these parameters. The situation is reminiscent to the simpler case of harmonic lattice dynamics where intra-ionic electron-ion forces (K 1 ,K2 ) may lead to rather long-range formal ion-ion forces (refer to Sect. 4).

From (18.7) to (18.11) we can obtain the following results: (1) At long wavelengths, q~O, the Raman scattering from intra-ionic polariz

abilities of anions may be formally expressed in terms of relative displacements between cations and anions, U~(1) - U~(2), including, at least, 2-nd nearest neighbors.

(2) If the cations are heavier than the anions (like in MgO, CaO, etc.), the cation displacements are small in the optic branches, U~(1) ~ U~(2). Then (18.7) and (18.11) show that the formal polarizabilities are given by a (possibly quite spread-out) series of coefficients. They describe the characteristic 2 LO part of these crystals which is not obtained by the nearest and 2-nd nearest neighbor coefficients only (refer to PASTERNAK et aI., 1974). A numerical evaluation of the parameters in the case of MgO (MIGONI and BILZ, 1976) shows that coupling coefficients up to 4-th nearest neighbors have to be considered. In addition, from this analysis it turns out that among the different overtones those near the critical points at Wand L contribute the most important part of the Raman intensity. The reason is that these are saddle points where the cations are at rest so that the displacements of the anions are maximized.

As mentioned in the introduction to this section the intra-ionic nonlinear polarizability of 0-- originates from the instability of the free ion. As a consequence, the 2p-wave functions of 0- - re-adjust themselves if the lattice constant a or, generally, the positions of the neighboring ions are changed. The changed wave function if developed into wave functions of the original configuration may be described by (refer to Fig. 18.3)

2p(ao+Llao)=c2 2p(aO)+c3 3p(ao)+ ... (18.12)

i.e. a series of (np) wave functions of 0- -. As the inspection of the band structure of MgO demonstrates (RABII and BILZ, 1976 unpublished), the corresponding (np) states belong to rather high-lying energies in the conduction bands, the lowest of them are: 3s(Mg+), 3s(0- -), 3p(Mg+), 3p(0- -), etc. However, the wave functions of the low-lying Mg+ bands, 3s and 3 p, exhibit a strong admixture of oxygen, 3s and 3d, wave functions. This means that the dipole transition matrix elements between the highest occupied oxygen 2p band and the first few conduction elements are probably dominated by contributions of the oxygen wave functions. This would finally bring the band picture into agreement with the local (model) picture and allow for a rather precise (microscopic) definition of intra-versus inter-ionic Raman scattering.

b) Other diatomic ionic crystals. Raman spectra of many other diatomic crystals have been measured. All oxides with NaCI-structure, such as FeO, NiO, etc., exhibit 2-nd order spectra in the 2 LO regime. The same holds for the chalcogenides of rare earth metals with oxygen or sulfur anions (EuO,


EuS, etc., refer to GONTHERODT et aI., 1976}. Here, the analogy with the alkali earth oxides seems to be obvious. The chalcogenides and pnictides with tetrahedral coordination (ZnS, GaAs, etc.) show a similar behavior but in this case a strong covalency of the bonding has to be considered which modifies the interpretation of the spectra. We shall, therefore, discuss them after the covalent crystals in Sect. 19b.

c) Perovskites. A particular interesting class of crystals are those with a pure or distorted perovskite structure, such as SrTi03 , KTa0 3 , etc. Many of them are ferroelectrica at lower temperatures and show one or several phase transitions whereby the phase above the highest transition temperature is often a cubic paraelectric one. This phases are the simplest for a theoretical analysis and, among them, SrTi03 has attracted the greatest interest since single crystals are easily available and detailed inelastic neutron scattering data are published (COWLEY, 1964; STIRLING, 1972; IIZUMI et aI., 1973). The controversial results of the two latter papers have led to a re-examination of the dispersion curves by STIRLING and CURRAT (1976) which have essentially confirmed STIRLING'S earlier data.

Since in this structure the cubic phase has inversion centers at each of the different ions only second-order spectra are allowed. Even in the distorted ferroelectric phases these spectra show very strong intensities comparable to those of the first order lines if oxygen is the anion while perovskitic halides such as KMnF3 , etc., exhibit no measurable 2-nd order spectra (refer to PERRY, 1973). It is therefore tempting to extend the intra-ionic treatment of the diatomic oxides to the case of oxydic perovskites.

The first calculation of the unpolarized spectra has been carried out by BRUCE (in STIRLING, 1972). He used a quartic dipolar and a cubic breathing polarizability of the oxygen ion as in his calculation for NaCI (BRUCE and COWLEY, 1973), and obtained satisfactory agreement with the experimental data. The use of a breathing deform ability, however, seems to be debatable since, as in the case of N aCI and SrO discussed in force going sections, no evidence for a breathing effect in the lattice vibrations of SrTi03 could be established (STIRLING, 1972). Instead, one might expect that the anisotropic position of the oxygen ion between two neighboring titanium ions in one direction and four strontium ions in the perpendicular plane should lead to an anisotropic polarizability of the oxygen both in lattice dynamics and in Raman scattering.

RIEDER et al. (1975) have measured the three components of the Raman spectra and have analysed the data in terms of a modified version of one of Stirling's shell model. This version contains a positive shell charge at the Sr ion (overlap shell model) and an anisotropic (elliptic) oxygen polarizability. The results indicate a strong anisotropy but were not very satisfactory with respect to the different positions of experimental and theoretical peaks. Recently, an extended and corrected treatment including the analysis of the KTa0 3 spectra has been published (MIGONI et aI., 1976). Since here only a few low-lying dispersion curves have been measured the agreement between theory and experiment is gratifying (Fig. 18.7}. It turns out that the harmonic and the

246

o

n tT,O-TAIXI ,I .. 15 :~2TAIX.M)

200


KTa 03

Alg

Eg

'--

400 600

frequency shill [cm-l )

8 ~ 0 0 ,: '" N N

I I

1200

Fig. 18.9. Second order Raman spectra of KTa0 3 . (After MIGONI et aI., 1976)

Sect. 18

quartic intra-ionic polarizability is strongest in the tantalum-oxygen direction which may be related to the strong hybridisation of the 2p(O- -) wave functions with the 5 d (Ta +) wave functions (cf. the similar case in the partially covalent IIVI and III-V compounds, Sect. 19b).

d) Other ionic crystals. Many further ionic crystals with three or more particles in a cell have been measured in recent years. Usually, no theoretical analysis is available except of a tentative critical point analysis which has to be taken with caution as long as no detailed lattice dynamical data are known. The lack of these data prevents very often a deeper theoretical understanding at the time being. In addition, the complexity of the spectra requires a theoretical effort which seems to be outside of the present computational possibilities.

e) Photoelasticity and Raman scattering.

1) Photoelastic constants. At long wavelength where the phonons may be treated in the elastic (Debye) approximation inelastic light scattering can be performed by using the technique of Brillouin scattering (refer to KRISHNAN,

1973). As mentioned in Sect. 13, the coupling parameters in this frequency regime are the photo-elastic constants which have much in common with the elastic constants since both represent a fourth-rank tensor. However, in addition


to the 21 independent elastic constants in a crystal of general symmetry, 15 more constants have to be considered for the photo-elastic tensor since its antisymmetric part which is related to acoustic shear waves does not disappear in optically anisotropic media (NELSON and LAX, 1970; LAX, 1974).

The photoelastic constants describe the change of crystals' polarizability, i.e. soo' under strain. The general relation between the strain tensor e (refer to (10.22)), the dielectric tensor Soo and the photoelastic tensor p reads (NYE, 1960)

LI(S-l)= p. e. (18.14)

We have dropped the subindex 00 in the following. Without applied strain the change of S -1 may be described by

[LI(B- 1Hxp= Llsap , Baa' Bpp

which means in cubic crystals with B being a scalar

LI(B- 1)= LI: . B

From (18.14) the photoelastic coefficients are

____ 1_. JBap Papyo- J'

Baa' Bpp eyo

We use the abbreviated suffixes (NYE, 1960) with

rxfJ 11

1 22 2

33 3

23,32 4

31,13 5

which gives 1 aBap ( ~)

Pik= - 2 -a- 1 +uyo • Baa' Bpp eyo

12,21 6

(18.15)

(18.16)

(18.17)

(18.18)

(18.19)

As one is working in the theory of elasticity with two sets of constants, the elastic stiffnesses C ik and the elastic compliances Sik' one uses sometimes instead of the elasto-optical or strain-optical coefficients Pik the stress-optical or piezooptical coefficients qik which are related to the Pik by the equations

Pik=qijCjk' (18.20)

Since Pik =\= Pki' in general, we have 6 x 6 independent coefficients, as stated above.

II) Photoelasticity of cubic ionic crystals. In the case of cubic crystals we have as for elastic constants only three photo elastic constants. i.e. Pl1' P12 and P44' With respect to cubic symmetry we may define in the same way as for the polarizabilities in Sect. 13, e.g., for a rocksalt structure, a hydrostatic coefficient

and two shear coefficients

p(rt) == t(Pl1 + 2 pd, (18.21)

p(rii)==1-(P11 -pd,

p(r2"5)==P44' (18.22)


A particular simple case is the change of the dielectric constant under hydrostatic pressure. This leads to change of the cell volume of the crystal, v, (10.51) and a similar change of the interionic distance, R, or the density p, which results in following strain derivative of e (MULLER, 1935):

p(Ft) = _~ R ~= _~ dIne = _~ dIne = +~ dIne. 3e2 dR 3dlnR edlnvo edlnp

(18.23)

a) The Lorentz-Lorenz limit. The strain derivative of e can be related to the ionic polarizabilities if the local field is known. In the point ion limit the local field is given by the Lorentz-Lorenz formula, (6.82) with j = 1, which gives for the change of e

dIne =~~=_(8+2)(e-1) (1- dIna) dlnvo 3e dR 3e 3dlnR

a dln-

= + (e+2)(e-1) __ v_o ~ _~ (1- dIna). 3e dlnvo 3 dlnvo

(18.24)

(18.25)

Obviously, if the ionic polarizabilities a + and a _, with a + + a _ ~ a, change very little with the interatomic distance, i.e. dIn a/d In Vo ~ 0, we have

dIne e ~->-dIn Vo ~ 3·

(18.26)

If, one the other hand, e changes dramatically with R, we may expect a relation such as d In e e d In a

dlnvo ;:53 dlnvo. (18.27)

The change of e with strain has been discussed by MUELLER (1935), BURSTEIN and SMITH (1948 a, b) and, more recently, by SHARMA et al. (1976). It turns out that for the alkali halides one is always in the negative regime of dIn e/d In Vo with values for dIn a/d In Vo between 0.4 (KI) and 0.7 (LiF) which indicates a change of aooR to R2. A long known exception is MgO where dIn e/d In Vo >0 and aooR4. This result was considered for a long time to be a strong puzzle but it follows rather directly from our discussion in Sect. 18aIII about the Raman scattering of the alkali earth oxides. The instability of the 0 - - ion as a free ion leads to a volume dependence of the ionic polarizability which is greater than the classical value aooR3. Let us concentrate, for the moment, on crystals such as LiF and MgO where the polarizability stems nearly exclusively from the anion and let us assume that this polarizability is proportional to a certain power of R:

( R )3(1 +~)

a>a ~ao - (18.28) ~ - - Ro '

a dln-

d In a Vo dlnR=3(1+1]) or dlnvo =1], (18.29)

8.p(F1+)=- dIne =_1](e+2)(e-1)~_~e.1]. dIn Vo 3e 3

(18.30)


This parameterization is rather convenient. It shows that the hydrostatic photoelastic coefficient p(I;+) should be positive for all crystals where the straindependence of the effective polarizability is smaller than the classical value, i.e. 1] < 0. This is the case for all alkali halides and many non-oxidic ionic crystals. In the case of oxides we shall generally expect that 1] > 0, and, if hybridization plays a role as in the case of spinnels or perovskites 1] may be close to 10 or even greater which should lead to strong negative values of p(rt) ~ -1]/3. A more precise treatment requires, however, a consideration of the deviation from the Lorentz-Lorenz limit.

/3) The Drude limit. If we allow for an Adler-Wiser correction factor y<l, (6.82), the strain derivative of 8 is, instead of (18.24),

(dln~ ) Rd8 [3+y(8-1)](8-1) __ v~_ 3(8-1) ~ . 8dR 8 dlnvo 3+Y(8-1)dlnvo

In the Drude limit, y-.O, dy/dR-.O, we have

rx dln-

~ ~ 3 (8 - 1) __ v_o = _ 8 - 1 (3 _ d In rx ) . 8 dR 8 dlnvo 8 dlnR

(18.31)

(18.32)

The essential difference of (18.32) as compared to (18.25) is the replacement of the factor (8 + 2) in (18.25) by 3 or, at high values of 8, the reduction of p(rt) ~ -1]/3 to -1]/8. That means that the enhancement of the photoelastic constant by an increasing exponent of the polarizability is more or less cancelled by the delocalization of the electronic charge which decreases the local-field effect drastically. A rough interpolation formula is

~ ~~ (1 + Y8) 1]= -p(r+)· 8 38 dR 3 l'

(18.33)

where the change of y has been neglected. It shows that even in the general case the qualitative behaviour of the strain derivative of 8 is given by the exponent 1] which is zero for the 'classical' case, rx ex: Vo'

Since Ph is a symmetrized hydrostatic first-order Raman-tensor it describes the squared first-order hydrostatic deformation potential i.e. a symmetrized firstorder electron-phonon coupling. 1] = 0, i.e. Ph(rt) = 0, means that this effective coupling vanishes, which is consistent with the picture that the polarizability changes as for a 'free' electron charge density in a given volume. Ph <0 is the 'normal' case with an attractive electron-ion coupling while p > ° describes a 'repulsive' coupling corresponding e.g. to the instability of the 0- - as a free ion which supports the delocalization of the electronic charge in oxides with increasing interionic distance. Even the sign of the photoelastic constant gives therefore an interesting information about the qualitative character of the local electron-phonon coupling in polarizable insulators.

y) Shell model of photo-elastic constants. Recently, SHARMA et al. (1976) have correlated the photo elastic constants with parameters of the shell model,


where the polarizability of the exchange charge (DICK and OVERHAUSER, 1958) was used to express the strain derivative of the individual ionic polarizabilities Ct±' with Ct~Ct+ +Ct_, in terms of the parameters of Born-Mayer potentials. It turns out that one obtains

dCt± 2D --~C(+---,

dR - p(Y± e) (18.34)

where D means the exchange charge, Y ± e the shell charges, and p the screening radius of the Born-Mayer potential. In most of the alkali halides Ct_ > Ct+, so that Eq. (18.34) leads then to dCt/dR > 0, i.e. to a partial compensation of the volume term in (18.32). The effect is, however, much more dramatic in MgO where dCt_/dR~ -9dCt+/dR. Although the shell model being based on the assumption of stable free ions is not valid in this case the result obtained by SHARMA et al. (1976), is rather satisfactory.

c'i) The effective gap approximation. In Sect. 6 we have seen that in the dielectric model theory (PHILLIPS, 1971) the polarizability may be related to an 'effective' (Penn) gap

Ct = const. E; 2,

d In Ct _ 2 d In Ep dlnR - - dlnR'

(18.35)

(18.36)

The volume exponent of Ep (c.f. Eq. (18.28)) is therefore -l (1 +1/), and should become strongly negative for high values of 1/. We may therefore expect that crystals with anomalously polarizable ions such as oxides exhibit a strong increase of Ep under hydrostatic pressure since dEp ~ - d In va = - 3 d lnR, and dlnEp/dlnp~!(1+1/)~1//2. In the ionic limit, however, where 0<1/<1, we expect that the volume exponent is very small and Ep rather insensitive against the application of external pressure. This agrees with the ionic limit of the dielectric model theory, where

(18.37)

with Ee and Eh the Coulomb and the homopolar contributions to the effective gap.

In the ionic limit, Eh <f: E e , and (VAN VECHTEN, 1968)

dlnEh dlnEe dlnR <f: dlnR ' (18.38)

which agrees with the above-given statement in the strongly ionic limit. We should note, that our discussion is in contrast to an assumption by PANTELIDES (1975) where an equation

8 - 1 = const. R 3 (18.39) has been used which leads to

R d8 3(8-1) --e- dR= + 8 >0, (18.40)


which disagrees with the results obtained for alkali halides even in sign (SHANKER et al., 1975).

From (18.30) we expect in alkali halides values for the photo elastic constant 3p(rn: -1]/2;£Pl1 +2P12;£ -1], where 8~2 has been used. Since 0>1]>-1 may be defined as an 'ionic' regime for 1], we have for this case O<Pll + 2P12 < 1. Experimental values from Brillouin scattering are (BENCKERT and BACKSTROM, 1973),

Pll + 2P12: 0.437 (NaCl), 0540 (KCI), and 0.571 (KCI),

which is consistent with a mean volume exponent 1] ~ 0.5 of the crystal pol arizability. For further values of the photoelastic constants in ionic crystals the reader is referred to Landolt-Bornstein (1979, Vol. 11) and CARDONA and Yu (1969).

III) Second order Raman scattering. Since the mayority of ionic crystals investigated so far have cubic symmetry, first-order Raman scattering is not allowed due to inversion symmetry and cannot be related to the photoelastic constants. The foregoing discussion allows, however, a discussion of the interrelation of second-order Raman scattering with photoelastic constants via the volume exponent of the polarizability.

From (18.28) follows that the second derivative of rx is given by

and

d2 ln rx d(1nR)2

3 (1 + 1])[3(1 + 1]) -1J

~ [~~] _(8+2)(8-1)(8 2 -8+2) 2~ 2 dR 8 dR - 83 1] -1] 8,

in the Lorentz-Lorenz limit. From (18.30) we see that

~ [~~] ~1] ~ ~= -1]8· 3p(rn· dR 8 dR 8 dR

(18.41)

(18.42)

(18.43)

The different orders of the strain derivative of the dielectric constant are therefore given by corresponding powers of the volume exponent 1] and the derivative of the polarizability by powers of (1 + 1]) . 1]n = 0 denotes for all orders n the case of approximately vanishing electron-phonon coupling of n-th order and defines an 'ionic' regime for 1]<0 and a 'polarization-dominated' regime for 1] > 0 which allows very strong second order Raman intensities OC 1]2 in crystals with perovskitic or spin ell structure where 1] may be of the order of 10 (refer to Sect. 18 b). Our discussion does, so far, not contain any homopolar contributions to b8 00 which we shall discuss in Sect. 19 for the predominantly homopolar crystals.

f) First-order Raman scattering. We have shifted discussion of first-order Raman scattering in ionic crystals to the end of this section since it plays only a minor role for the understanding of the electron-phonon interaction in strongly ionic crystals. In contrast to the first-order infrared processes which define the infrared light absorption in terms of resonant dispersion oscillators the Raman


first-order phonon light scattering at lower laser frequencies is usually an offresonant phenomena which is forbidden in all crystals with inversion symmetry at the ionic lattice sites. In crystals with lower symmetry (CaF2' Ti0 2 , etc.) we can compare the frequencies of the first order Raman lines with those obtained by other methods ore, more important, with theoretical frequencies as obtained from a model or microscopic model. In addition, symmetry properties of the Raman lines may be checked and used for further clarification of the situation. Of much greater interest would be the determination of absolute intensities or of relative ones between lines of the same or (more difficult) different crystals. Very little is known about these parameters in strongly ionic crystals (refer to GUNTHERODT and CARDONA, 1982). We remark that a corresponding analysis is, presently, possible for some of the more homopolar crystals (refer to BIERMANN, 1974b; HAYES and LOUDON, 1978).

The same holds more or less for the field-induced first-order Raman scattering. Here, the importance of the method relies upon the possibility to determine frequencies of modes which are 'silent' in a normal Raman scattering process and only very roughly known from infrared absorption. The general representation of the field induced Raman polarizability has been given in Sect. 13. For experimental results and further details the reader is refered to BURSTEIN et al. (1974) and ANASTAS SAKIS (1980).

19. Raman spectra of covalent and partially ionic crystals. The first attempt to analyse the Raman spectra of a covalent crystal was by H. SMITH (1947) who tried to describe the unpolarized spectra of diamond as measured by KRISHNAN (1947) in terms of formal expansion parameters of the non-linear polarizabilities analogous to the treatment of BORN and BRADBURN (1947) for the spectra of ionic crystals (refer to Sect. 16). As in this case, Raman spectra are often analysed in terms of critical points of the one- and two-phonon density of states where the proper selection rules for Raman scattering in a tetrahedral crystal are considered. A collection of important selection rules are given in Table 40.11 For a detailed discussion we refer to BIRMAN (1974). Here, we are trying to discuss the situation from a model or microscopic point of view.

a) Spectra of diamond and its homologues. In Sect. 16 we have discussed the infrared properties of covalent elements C, Si and Ge. It was shown that the dynamical effect of the covalency of these materials could be represented by introducing a 'bond charge' (PHILLIPS, 1968; MARTIN, 1969) which was allowed to follow the ionic displacements adiabatically (WEBER, 1975). The second order spectra of infrared absorption turned out to be described with very few anharmonic coupling parameters between the bond charge and its neighboring ions. This demonstrates the close analogy between a dynamical treatment of the bond charge and the properties of an effective ionic charge in ionic crystals.

The situation is different for the polarizability and its change in Raman scattering in covalent crystals (Go et aI., 1975). This exhibits a one-phonon line with symmetry r2+s and two-phonon spectra of symmetry r1+, r1i and r2+s. The second order Raman tensor is given by (Stokes part) (13.7):

iaBvo(Q)=2 I Pa~()LIJ2)Pyo(Jl )'2)(1 +n1)(1 +n2) b(Q-Wl -w2)· (19.1)

Sect. 19 Raman spectra of covalent and partially ionic crystals 253

The polarizability tensor P may be expanded in powers of the ionic and bondcharge displacements in ordinary space as was done for the nonlinear dipole moment M(2) by Go et aI. (1975) (refer to Sect. 16):

p=pO+p1 +tP2: (U+ U+ + U- U-). (19.2)

The contribution of a cubic potential V 3 to the Raman spectra analogous to (16.5) for M(2) cannot account for the second-order Raman scattering since it leads to a very weak change of the crystal polarizability. Instead, we follow the concept of bond polarizabilities (BP) (WOLKENSTEIN, 1941; MARADUDIN and BURSTEIN, 1967; FLYTZANIS and DUCUING, 1969, 1972) and represent the polarizability of the covalent crystal P by a sum of independent BP's each of which is given by (WEBER et aI., 1974):

P~P {Rb} = L [R~R~all (Rb) +(bap - R~R~) a1.(Rb)] , Rb=Rb/Rb, (19.3) b

where all and a1. are the radial and lateral components of the BP. The expansion coefficients of P defined in (19.2) are then simple linear combinations of a II' a 1.' and their derivatives:

4 4 aV =-3-(a li +2a1.)' aq=-3-(all-a1.)' a1 =roa~, Qo Qo

a'l =r~a~, a25 =roaq[ln(aq/(Rbf)]', (19.4)

r 0 is the equilibrium bond length. Thus

o Qo ~P = Q o av ba/3 = 4n (8 00 -1) ba/3'

~L(r;1) = ~c;.; a25 8ap" ro V 3

~;"v(r;.+) = 1~~6 ( - a'l + (1- 3 b".) ( 1)

Q ~~"v(r;.1) = 24~6 (2a 25 + 3 aq b"v)(2 - sa/3,,- sapv)

~~"v(r;1) = Q2 02 (- a~5 - 3 a25 (b"v + 1-8aPv -8a/3) 1 ro

-9aq(1-b"v)(1-8~p,,-8~/3v))' (19.5)

where the subindices a, {J refer to the Cartesian coordinates of the dielectric tensor and its derivatives; fl, v are the coordinates of the differences in the displacement of near-neighbor atoms; 8ap fl' the Levi-Civita tensor, is given by

8~/3" = (1- bap)(1- ball)'

sa/3" =(1- ba/3)(I- ba,,)(I- bp,,). (19.6)

Calculations of the first- and second-order Raman spectra of C, Si, and Ge were carried out with the five fitting parameters aq , a1, a'l' a25 and a~5. av was


15

OL-~ ____ L-__ L-__ ~ __ ~~~

2200 2400 2600 Q [em-I]

Fig. 19.1. Second-order Raman spectra of diamond at 300 K. The solid lines are the experimental spectra of SoLIN and RAMDAS (1970). The histograms are calculated spectra (Go et aI., 1975a)

obtained from Goo. The dominant component of the experimental spectra is that of .r;.+ symmetry accompanied by a weaker I;.~ quadrupole and a very weak .r;.1 scattering. These facts are well reproduced by the model in all three substances.

In Fig. (19.1) the three spectra of diamond are compared with calculations by Go et al. (1975 a). The two prominent spectra with .r;.+ and I;.~ symmetry are practically determined by 0c'l and OC~5' respectively, which are approximately equal. This corresponds to a very low value of OC.l as compared to oc II in diamond (ocq ~ ocv) and indicates a simple (radial) charge-transfer mechanism in this crystal. Since the intensity of the measured Raman spectra is only known in arbitrary units Go et al. determined ocq from the photo elastic constants P11 +2P12' The nonfitted photo elastic constants Pll -P12' P44 agree for diamond very well with experimental values (Table 19.1).

A feature of particular interest is the peak at the cutoff of the two-phonon spectra which often is attributed to a two-phonon bound state (Ruv ALDS and ZAWADOWSKI, 1970). Since the calculation represents well the frequency of the experimental peak (slightly above 2WRaman) as well as its shape it seems that the result does explain the two-phonon spectrum of diamond without invoking a bound state. A certain "overbending" of the LO mode in the (100) direction above the Raman frequency with a subsequent extension of the density of states in this frequency region was first proposed by MUSGRAVE and POPLE (1962). Thus within this model the peak would be due to an overtone volume scattering,


Table 19.1. Values of quantities of Raman scattering in diamond as calculated by Go et al. (1975a). The experimental values are given in brackets

Boo

IXI/IXq

IX~/IXq IX25/IX.

IX~5/IX. Pll +2P12

Pll - Pl2

P44

IXv

IX.

IX II (A3) IX1- (A3) pel) (A2)

IdI/

" LEIGH and SZIGETI (1970) b BIEGELSEN (1974)

c

(5.86) 4.13

284.65 2.13

255.6 -0.16

( -0.16)" -0.283

( -0.293)"

-0.172 ( -0.172)"

0.387 0.387 3.293 0.0 3.5

-1.82-(±4) 334

(250)

C TRAPPENIERS and VETTER (1972) d CARDONA (1973) _ SWANSON and MARADUDIN (1970)

Si

(11.7) -46.16

-180.D2 -23.08

0.0 -0.058

( -0.058)b 0.013

( -0.167)"

-0.0076 ( -0.082)"

0.851 -0.069

6.006 7.749

13.95 6.73-

93.4 (35)

f Ratio of the first- to second-order Raman intensity

Ge

(16.3) -57.45

-288.22 -24.39

-248.78 -0.28

( -0.28)C 0.016

(-0.0095)" (O.Oll)d 0.019

( -0.074)" (0.012)d 1.218

-0.167 9.89

15.49 43.1 40.53-

52 (60)

in agreement with the conjecture of UCHINOKURA, SEKINE, and MATSUURA (1974) and a recent density of states analysis by TUBINO and BIRMAN (1975).

For germanium and silicon a large increase in ()(.L is observed (Table 19.1). The calculated first-order Raman tensor agrees quite well with that calculated by SWANSON and MARADUDIN (1970), while for diamond the value has the opposite sign. This discrepancy might be due to the strong dependence of the Raman tensor on b in Swanson and Maradudin's calculation. We note that the Pll - P12 and P44 calculated with the model for germanium have opposite signs compared to those observed experimentally. This rather disturbing fact is a consequence of our assumption of simple bond polarizabilities which is good for diamond. For germanium, however, it is known (CARDONA, 1973) that two main mechanisms contribute to the polarization: the average band gap E2 ('Penn' gap) and the lowest direct gap Eo. While BO() is determined mainly by E 2 ,

in the differential parameters one encounters equal contributions of both mechanisms which can even have opposite sign (HIGGINBOTHAM et at, 1969). The shear photo elastic constants, determined at long wavelength by Eo and E 2 ,

reverse sign near Eo because of the increasing contribution of this gap. The first-


order Raman tensor of germanium, however, is mainly determined by Eo and no antiresonance cancellation occurs. P 11 + 2 P 12 is determined for long wavelength by E2 and so should be the r;+ scattering (CARDONA, 1973). Within this model, and assuming that the quadrupolar (~i and 1;1) scattering is produced mainly by the Eo gap, a fact which seems to agree with experiment, one may conclude that the calculated PI! - P12 and P44 represent only the Eo contribution to these quantities. In Table 19.1 the contribution of Eo to P11 - P12 as extracted by CARDONA (1973) is listed. The proximity of these contributions to the values calculated with the BP model is quite satisfactory. For a recent discussion of the photo-elastic constants in diamond we refer to GRIMSDITCH and RAMDAS (1975).

In Fig. 19.2 the experimental (HATHAWAY, 1973) and theoretical (WEBER et aI., 1975) spectra of silicon are shown. Similar results are obtained for germanium (Go, 1975). The interpretation of the photo-elastic constants (Table 19.1) is here more difficult than in the simple case of diamond but follows the same lines of argumentation. A deeper understanding of the situation may be obtained by focussing attention to the relation between the model parameters and the important features in the energy band structure (Go et aI., 1975b). We recall that the model assumes that the electronic polarizability of the diamondlike crystals can be expressed as a sum of the electronic polarizabilities of the bonds between nearest neighbor pairs of atoms, each of which depends on the bond length only. Furthermore it is assumed that the bonds possess axial

VI C

Si 305 K

10

8 1)

'" 2 .~ c c E c

c::

400

.. I " " "

)~ 600 800 1000

Frequency (cm-I )

Fig. 19.2. Raman spectra of silicon at 305 K. Experimental curves: dashed lines (TEMPLE and HATHAWAY, 1973). Theoretical curves: solid lines (WEBER et a!., 1975)

Sect. 19 - Raman spectra of covalent and partially ionic crystals 257

symmetry with respect to the principal axis of the bond which is preserved throughout the nuclear motion. This model contains six parameters, which describe the electronic polarizability of the crystal up to the second order in nuclear displacements. Firstly, we discuss the connection between the model and the energy bands using the tight binding approximation. The wave functions of valence and conduction bands of diamond-like crystals can be expanded in terms of Bloch sums constructed from the sand p atomic orbitals at each atomic

site: p"n=N-1/2I~,>ikRiA~ (~)u/r-R;). (19.7) j Ri J

The static dielectric constant is given In the Hartree approximation by the equation

B (0 0) = 1 + 8n e2 '" < l/I~ I rail/lD < l/I~ I rpll/lD .

"p , V L.. Ee _ EV v,e,k k k

(19.8)

In the simple bonding-antibonding picture the energy denominator is replaced by the energy difference between the bonding and antibonding states and thus can be taken out of the summation. The sum is then given by the dipolar matrix element between the bonding and antibonding s p3-hybrid orbitals which are localized along the four bond directions and lead to a radial bond polarizability, 0(11' only. This situation has been found in diamond. On the other hand the non-zero lateral bond polarizability, 0(1., in Si and Ge indicates a breakdown of the bonding-antibonding picture for these materials. The essential trend in the band structures when going from diamond to germanium is the drastic lowering of the r; conduction band which results in an increasing contribution of transitions between the highest valence band r;s and this r; conduction band to B"p(O, 0). The wavefunctions of the r;s valence band are essentially given by two pn and one pb bonding type atomic orbitals of the nearest neighbor pair of atoms while those of the r; conduction band are mainly given by the antibonding type of s atomic orbitals. The contribution via the Eo gap to 8 by the matrix element between the bonding pn and the antibonding s functions corresponds to a lateral bond polarizability. As a consequence the lowering of the Eo gap gives to an increasing of the 0(1.'

To show the relation between the bond polarizabilities and the band structure in more detail it is convenient to use binding orbitals directed along the bond directions, so called bond orbitals. Without specifying the bond orbitals two orthogonalized Bloch functions for the ground states and excited states may be constructed:

l/I~ = N- 1/2 L I eikRi V: (v) 4>b(r - Ri )

b Ri

(19.9) b Ri

b denotes the four bonds in an unit cell, 4>b and CfJb are the bond orbitals in the ground states and the excited states, respectively. Substituting (19.9) into the expression for B"p(O, 0) we obtain

0(11 =(11 +I2)1I12+212~2

0( 1. = (1 1 + 12 ) P} - 2 12111 P1.

258

with

and


11 =2e2 L A;v1(k)/(E~-E~) c.v.k

12 = (2/3) e2 I A;; (k)/(E~ - En c,v,k

A~~' (k) = V,t(v) C~(c) q'*(c) V,:" (v)

111.1. =<4\1 rL l€pb)'

Sect. 19

(19.10)

Again Il( 1. becomes zero if we replace cI>b and €Pb by bonding and antibonding s p3 orbitals. For the Eo gap, cI>b is a pn bonding type and €Pb a s antibonding type orbital and therefore the matrix element ~ is different from zero, so Il( 1.' Il( 1. even becomes more important as the Eo gap decreases.

F or an a b initio calculation of the bond polarizabilities a variational perturbation procedure can be used as proposed by FLYTZANIS (1969). Using only the ground state Slater type s p3 bond orbitals (which in the lowest approximation corresponds to a closure approximation) Go et aI. are able to find a systematic increase of Il( 1. relative to Il( II from diamond to germanium. However, a more accurate wave function seems to be necessary in order to make a comparison with the fitted values of Il( II and Il( 1.'

It has to be noted that the consideration of local field corrections (HANKE • and SHAM, 1974) results in a quantitative modification of the results without changing the qualitative picture.

One should remark that TUBINO et al. (1975) have independently derived similar results using the same concept of bond polarizabilities. In particular, the peak at the two-phonon cut off observed in diamond has also been attributed by these authors to overtone scattering. Their work differs from that by Go et aI. by using only the zero and first order derivatives of the bond polarizabilities. One notes that COWLEY (1965) in his first attempt to calculate the Raman spectra of diamond used only one parameter which is equivalent to the second derivative of the longitudinal bond polarizability and obtained a rather good result for the unpolarized Raman spectrum which mainly consists of r;. scattering.

b) Spectra of III-V and II-VI compounds. Raman spectra of III-V and II-RI compounds have been investigated by many authors. Here we concentrate on cases where, at least partially, an analysis in terms of model or microscopic entities has been undertaken. For further information refer to CARDONA (1981).

We start by discussing first the III-V compounds, in particular the gallium pnictides which belong to the best investigated semiconductors. The lattice dynamics of GaAs has been successfully calculated by WEBER et aI. (1974) using the bond charge model, where the bond charge is moved away from the midpoint between Ga and As ions towards the As ions. Using the same model as for homopolar crystals Go et aI. (1975) have calculated the three second order Raman spectra of GaAs, The comparison with the measured spectra (TROMMER et aI., 1975) is shown in Fig. 19.3. With respect to the good agreement obtained between the calculated and the measured spectra it seems


Ii GaAs 370 K

(/)

5 ~ ...... z: ::J 4

en 3 cr: c:(

~2

>-~ ...... (/)

z: w 0 ~ 2 z:

z: c:( ::E

0 c:( cr:

Fig. 19.3. Seond-order Raman spectra of GaAs at 370 K. Solid lines: experimental spectra (TROMMER et aI., 1976), histograms: calculated spectra (Go et aI., 1976). First order TO and LO lines are

not calculated

that the bond polarizability model gives a reliable description of the polarization properties of mainly covalent crystals.

For the strongly ionic II-VI compounds like ZnO and ZnS an extension of the bond charge model seems to be very difficult (RUSTAGI, private communication). Very recently, it was shown (KUNC et BILZ, 1976) that a shell model including valence forces is well able to describe the lattice vibrations and the second order Raman spectra of all zinc chalcogenides. Here, only intra-ionic nonlinear polarizabilities have been used. This result indicates that the bond charge model in our discussion may be limited to the predominantly covalent IV - IV and III -V compounds. Because of the general interest in the question of the applicability of an 'ionic' model to the case of partially covalent crystals we follow the treatment of KUNC and BILZ some what more in detail. They start out by noting that the electronic structure of the semiconducting compounds, such as the tetrahedral II -VI and III -V crystals is presently quite well understood. The principal features of their valence and conduction bands can be analysed with a relatively small number of tight binding matrix elements and a complete trend analysis of the valence bands from the ionic alkali halides up to the covalent diamond type crystals has recently been published (PANTELIDES and HARRISON, 1975). On the other hand, the dynamical properties of those crystals, as seen in phonon dispersion curves and Raman spectra are still waiting for a physical interpretation.

KUNC and BILZ attempt to analyse the dynamical properties of the III -V and II -VI compounds and to demonstrate the intermediate role of these crystals between the ionic and the covalent crystals. In addition they try to elucidate some essential properties of these limiting cases which seem to be at present not


well understood. They start from the observation that a wide variety of binary compounds with rocksalt structure can be described with the help of an overlap shell model (OSM) (BILZ et aI., 1975; refer to Sect. 5). Going to the zincblende structure one remembers many attempts to describe the phonons in II-VI and III-V compounds with the help of rather elaborate shell models without an understanding of the internal connection between the different models for those crystals except for a relatively successful application of a deformation dipole model (KUNC et aI., 1975). Analysing the different calculations in the literature one finds that two of the best models for ZnS and ZnTe (VAGELATOS et aI., 1974) are in fact OSM's where the shortrange parameters are expressed in terms of valence forces. By extrapolating and interpolating the numerical values of parameters of these two models one obtains a 'valence-overlap shell model' (VOSM) for all chalcogenides and Ga pnictides (GaP to GaSb). After a slight readjustment, these models give good results for all crystals which means that the parameters generally show rather systematic trends and small changes. The static ionic charge is known to be in the rocksalt structure about 0.9 to 1 e for alkali halides and 1.7 e to 2 e for the alkaline earth oxides. This charge turned out to be about 1.9 e for the zinc chalcogenides and about 2 e for the gallium pnictides. These values differ strongly from the values obtained by extended shell models, particularly for the gallium pnictides (refer to PRICE et aI., 1971).

The investigation of the Raman spectra in these crystals recalls that the Raman spectra of the cubic oxides which behave in this respect like open-shell crystals, can be successfully described mainly with intra-ionic non-linear polarizabilities (refer to Sect. 16). It seems, therefore, to be appealing to attempt a similar analysis of the tetrahedral II -VI and III -V compounds, which are partially ionic systems with increasing covalency. Kunc and Bilz introduced non-linear polarizabilities such as used by BRUCE and COWLEY and by BuCHANAN et aI. for the ionic crystals (refer to Sect. 16). The intraionic part of the non-linear shell model potential reads up to fourth order in the relative shellcore displacements w(l K)

c[JNL= L {H l(K) Iw;(IK)w~(lK)+H2(K) I W:(lK) II< ap a

+H4(K) L ICapyl wa(lK)wp(IK)w/lK)}. (19.11) apy

This potential leads in a straigh'tforward way to the differential polarizabilities ~p(Al ):2) which determine the Raman cross section (refer to (16)).

The concept of intra-ionic polarizabilities means that the ions contribute individually to the spectra. Generally, one may expect that lighter or more polarizable ions show stronger scattering than heavier or less polarizable ions. The second order Raman cross section is proportional to the fourth power of the core displacements i.e., in the harmonic approximation, to the inverse square of ionic masses. The mass factors appear explicitely in the polarizabilities ~p(Al; ):2) and dominate over the influence of ionic polarizabilities. We note that the heavier atom is responsible for the lower-frequency part of the spectra, the lighter for the higher frequency part.


LO

ZnS

o 100 200 300 400 500 600 700 Wavenumber I cm-1

Fig. 19.4. Second-order Raman spectra of ZnS. Solid lines: Experimental data (KRAUZMAN, Thesis 1969, unpublished). Histogram: Calculated spectra (KUNC and BILZ 1976). First order TO and LO

lines are not calculated

From this analysis it turns out that the spherical quartic part of the nonlinear potential, (H1 in (19.11)) leads to a satisfactory description of all second order Raman spectra. The inclusion of the 11 to the T1S intensities gives no significant improvement within the accuracy of the calculations and has, therefore, been omitted. The same holds for the cubic term (H 4) responsible for the first order scattering. Its contribution to the second order spectra (H 4)2 is inconsistent with the observed intensity ratios between first- and second order scattering since it underestimates the Tl part of the scattering by several orders of magnitude. We are left with the fitted coupling parameters H 1 • They show a systematic decrease with increasing size of the anions. The highest value appears for the oxygen anions, which reflects the fact that ZnO exhibits a very strong 2 LO band above 1000 cm -1 similar to the corresponding bands in oxides with rocksalt structure (refer to Sect. 18b). The anomalous high scattering efficiency of the oxygen ion originates from the strong variation of its polariz-

262 Lattices with point defects Sect. 20

ability with the ionic radius (TESSMANN et a1., 1953). As an example, the results for ZnS are shown in Fig. 19.4. Serious deviations of the theoretical curves from the experimental data are indicated only in the low frequency acoustic regime of the spectra. Introducing many more parameters into the model would improve the calculated spectra but would, on the other hand, destroy the simplicity of the approach.

It should be noted that the spectra of GaAs are measured above resonance while those of GaP are still strongly influenced by resonance effects which cannot be reproduced in this approach. We also note that N in GaN does not show the anomalous high Raman efficiency of 0 in ZnO (R. Tsu, unpublished).

A comparison of this treatment with the bond polarizability approach discussed in the foregoing section shows that both models contain a longitudinal bond polarizability which is equal to the sum of the two intra-ionic polarizabilities for equal ions. The statement that the second derivative of the longitudinal bond polarizability governs the situation and leads to overtone spectra may be shown to be equivalent to the importance of the quartic (H 1) term.

We mention that the bond charge model is missing any correlations between different bonds. This causes the above mentioned difficulties for an extension of this model to the more ionic crystals such as the II -VI compounds. The shell model on the contrary, contains bond-bond correlations at the very beginning since a shell displacement at the ion automatically affects all four bonds connected with that ion. This may be a reason why the shell model with intraionic differential polarizabilities works for the tetrahedral II-VI compounds and even for the rocksalt-type oxides. In this respect the oxygen behaves more like an open-shell ion even in the rocksalt structure. It can, therefore, not be compared with the closed-shell ions in alkali halides where the ionic radius seems to be a well-defined constant. The corresponding rigidity of the charge density distribution around the ions may cause the fact that inter-ionic differential polarizabilities govern the Raman spectra of the alkali halides (KRAUZMAN, 1973 refer to Sect. 18a). It means that for these crystals a change in the polarizability requires a charge transfer between cations and anions.

We arrive at the conclusion that a simple shell model is able to describe the trends in the lattice dynamics and Raman spectra from the ionic to the covalent crystals mainly in terms of intra-ionic linear and non-linear polarizabilities. This may provide a basis for a further detailed comparison with the corresponding trends in the band structures.

F. Lattices with point defects

20. Types of defect and their effects.

a) Introductory remarks. In this chapter the general theory, which was developed previously for perfect crystals, is applied to perturbed systems. In order to make this chapter more or less self contained the repetition of results from other chapters cannot be avoided.

Sect. 20 Types of defect and their effects 263

The study of defect-induced spectra is hoped to lead to an understanding of the coupling of a defect to the lattice. The defect often has the property of lowering the symmetry of the lattice, thus allowing excitation, scattering, decay, and other processes which otherwise would be forbidden for reasons of symmetry. These defect-induced processes convey very detailed information about the host crystal, so that the study of defect-induced spectra is a useful tool for the investigation of the properties of perfect crystals (Sect. 21).

The theory will be developed mainly for crystals with simple defects since this is the easiest way to demonstrate the principles according to which the general theory of pure crystals has to be applied and modified, and since the results so obtained can be compared with correspondingly simple experimental data. We will again restrict ourselves to dielectric crystals, defects in magnetic crystals are reviewed by COWLEY and BUYERS (1972).

Figure 20.1 shows schematically various types of defects in a two-dimensional crystal lattice. We next describe these various types of defects and their effects and define what we mean by "simple" defects (Sect. 20). Most of the defects shown in Fig. 20.1 will be given a more or less rigorous harmonic latticedynamical treatment.

Before one can understand infrared and Raman spectra due to defects, one needs to understand the dynamics of defective lattices. For the reasons mentioned above, the theory will be developed first for a point-ion crystal with one simple point-ion defect. The equations of motion of the defective lattice will be given in Sect. 22. The frequency dependence of the infrared and Raman spectra will be expressed in terms of the Green function of the defective lattice, which in turn will be developed from the pure lattice Green function in Sect. 23. Several important properties of the Green function will be discussed in Sect. 24. Expressions for the infrared and Raman spectra will be derived in Sect. 25.

After the theory for the simple point-ion defect has been developed, various generalizations will be made, the first being for molecular defects (Sect. 26). These defects have internal vibrations. These vibrations, which can to some extent be regarded as Einstein oscillations, cause characteristic spectra.

Another type of internal degree of freedom is that of deformation (polarization, breathing, etc.) of the electronic cloud around an ion or atom, which has been shown to be an essential feature in the understanding of lattice vibrations (Sect. 4). The results will therefore be generalized to a deformable defect in a lattice with deformable ions. A simple explanation of higher-order dipole moments by means of the shell model can be given; finally, it will be shown that a model interpretation of the Raman tensor will give suggestions as to which electronic excitations may be important around a defect ion (Sect. 27, see also Sects. 4 and 6).

So far it has been assumed that there is only one defect in the crystal. Of course, this does not apply to real crystals. In fact, a mixed crystal contains a large number of "defects". Because of the recent interest in amorphous, i.e. highly imperfect substances, this field is in rapid progress. Most existing theoretical results are restricted to low concentrations because of either analytic or numerical complexities, or both (Sect. 28).

Mixed crystals offer a wide field for theoretical and experimental study of the


effects of high concentrations of defects. Although theoretical results are limited, the experimental data will be summarized in a short section (Sect. 29).

Finally, anharmonic properties give further information about the defectlattice potential. Anharmonicity is important in temperature-dependent phenomena. The self-energy of resonance phonons will be discussed in Sect. 30. Thermal expansion and pressure-induced frequency shifts can be treated theoretically in a similar way; this will be done in Sect. 31, where bulk and local elastic constants of an impure lattice will also be found.

We will try to elucidate the theory by simple numerical examples and by experimental data. References to the original literature will not be comprehensive, and the reader is advised to consult the existing review literature, too.

Some aspects and problems of the dynamics are common to both pure crystals and crystals with defects; these are pointed out in some places in the text.

General treatments of the effects of defects may be found in several reviews (LIFSHITZ, 1956; LUDWIG, 1964; LIFSHITZ and KOSEVICH, 1966; MARADUDIN, 1965, 1966a, b; ELLIOTT, 1966; GENZEL, 1967, 1969; KLEIN, 1968b; SIEVERS, 1969,1971; NEWMAN, 1973; BAUERLE, 1973; BARKER and SIEVERS, 1975; TAYLOR, 1975; see also LUDWIG, 1967; and MARADUDIN et aI., 1972). Special attention to defects in metals has been paid in the review by LEIBFRIED and BREUER (1978) and DEDERICHS and ZELLER (1980).

b) Point defects, vacancies. The simplest possible defect in a crystal is the substitutional isotopic defect in an otherwise isotopically pure crystal. An example is the substitution of a CI- ion with mass 35 in an NaCI crystal with CI ions of mass 37. This defect merely causes a change of mass. It does not change the internuclear potential as long as one stays within the adiabatic and harmonic approximation.

Isoelectronic defects in crystals with the zincblende and wurtzite structure, except for the change in mass, do not seem to be accompanied by any appreciable change in their respective coupling to the lattice (AN GRESS et aI., 1965a; GOODWIN and SMITH, 1965; NEWMAN and WILLIS, 1965; GOVINDARAJAN and HARIDASAN, 1969; GAUR et aI., 1971; for example). They are, therefore, simple defects in the above sense, too, and are often referred to as mass or "isotopic" defects. The force-constant changes necessary to explain the localized-mode frequencies of defects of various valency in (partially) homopolar crystals were calculated by GRIMM (1975) and BELLOMONTE (1977 a), and systematic trends were found (see also LAITHWAITE et al. (1975); there are numerous investigations on particular defect systems carried out by the Indian School).

The U center (HILSCH and POHL, 1936, 1938) - a hydride (H-) or deuteride (D -) ion substituting for a halide ion in alkali halides - was the first defect whose influence on the lattice dynamics was studied by infrared techniques (SCHAEFER, 1960), and it has continued to attract the interest of physicists ever since. Unlike the previously mentioned mass defects, the U center does induce a local weakening of the repulsive short-range force constants by typically 50 % (FIESCHI et aI., 1964, 1965; JASWAL, 1965a, b; TIMUSK and KLEIN, 1966; XINH,


1966; PAGE and DICK, 1967; and others). In fact, implantation of non-isotopic defects into alkali-halide crystals generally does cause sizable changes in the internuclear potential, thereby inducing some distortion (Sect. 22b) in its immediate environment. This kind of defect is schematically shown in Fig. 20.1 a.

Changes in the potential (and hence the force constants) induced by a defect are generally assumed to be limited to a relatively small region near the defect. However, defects whose valency is different from that of the replaced host atom introduce charge changes into the crystal. These finally lead to Coulomb potential changes, which extend far into the lattice. Perturbations of this type have not been treated rigorously so far. The vacancy - a missing ion or atom as indicated in Fig. 20.1 b - is of this type except in group-IV crystals. However, care has to be used with respect to free-carrier absorption or further defects which, in general, have to be present for reasons of electrical charge neutrality. Changes in polarizability are related but can be treated (Sect. 27).

In the following a defect will be considered to be a "simple" one, if the crystal symmetry is not reduced beyond the point symmetry of a crystal lattice site, i.e., as long as the defect under consideration is located at a regular lattice site. Generalization to more complicated impurities will be made in due place.

a b

c d e

t-- H 9

Fig. 20.la-g. Various types of defects illustrated in a two-dimensional lattice: (a) Substitutional point defect; (b) vacancy; (c) substitutional off-center defect; (d) molecular defect; (e) interstitial; (f)

dislocation (point); (g) surface


Before going into mathematical details, we will first describe some dynamical properties of defects which are responsible for the characteristics of infrared and Raman spectra as well as the origin of these spectra.

c) Defect-induced infrared and Raman spectra. The discussion of defectinduced spectra has three aspects. First, there is the reduction of the symmetry of the lattice caused by the defect. Secondly, the defect may carry a different charge and may induce polarizability changes and thus may change the coupling of the phonon to light. Finally, the dynamics of the phonons to be excited is changed due to the presence of the defect.

In ideal crystals the periodicity of the lattice demands (quasi-)momentum conservation for interactions between various kinds of excitations in the crystal. Due to loss of the periodicity by introduction of a defect, this selection rule breaks down. The selection rule now only requires that the excited phonons have the appropriate representation of the point group of the defective lattice. For example, the combination of phonons which absorb infrared radiation has to have the symmetry of the light. In cubic crystals with inversion symmetry (point group 0h) the appropriate phonon symmetry for absorption is J{s(T1u),

while in Raman-scattering processes any of the even parity representations J{+(A 1g), J{i(Eg), and I;~(T2g) are allowed (LOUDON, 1964a). More details and some examples will be found in Sect. 22d. Various common notations for the irreducible representations of the groups 0h and Td as compiled by SLATER (1963) are given in Table 40.2 for the convenience of the reader who is familiar with notations other than that of LOUDON (1964a) which is used here.

As mentioned above the selection rules in perturbed crystals are less restrictive than in pure ones. In particular, for optical processes, this has the consequence that phonons with (resultant) wave vector other than zero can also be excited by external radiation fields. For a defect, whose perturbation is negligible and whose main effect is to lower the crystal symmetry, one expects an absorption spectrum which reflects the density of states properly weighted by the wave vector dependence of the coupling matrix elements and other factors. In fact, examples of this kind will be shown in Sect. 21. The reduction of the symmetry due to defects leads in principle to a change of one-phonon and a somewhat less pronounced change of multi-phonon spectra.

First-order Raman scattering is not allowed at all in those crystals in which all lattice particles are on sites with inversion symmetry. Introducing an impurity will leave only the impurity at a center of inversion. If the Raman scattering cross section is sufficiently large, defect-induced first-order Raman scattering is observed.

Analogously, the excitation of single phonons by infrared radiation fields in ideal crystals is restricted to the Reststrahlen oscillators (if there are any). Higher-order processes cause absorption over the whole frequency range. The defect-induced absorption is expected, in first order, to be proportional to the concentration of defects. Since the defect concentration usually is small (except in mixed crystals, see Sect. 29), the defect-induced processes are barely observable with today's techniques in frequency regions where there is appreciable host lattice absorption, unless localized modes or resonance modes occur (see Sects. 20d and 20e below).


Multi-phonon spectra are in principle much weaker than one-phonon spectra. Thus, in frequency regions where there is no one-phonon absorption in perfect crystals, defect-induced one-phonon spectra can be detected with rather high precision. Similarly, in regions beyond the pure-crystal second-order spectra, defect-induced two-phonon spectra can also be seen if they involve local modes and thus occur beyond the pure-crystal two-phonon frequency regime.

While the existence of defect-induced spectra is based on symmetry arguments, their intensity is governed by the phonon dynamics and by the coupling constants between the exciting field and the phonons. For example, defectinduced one-phonon infrared spectra in group-IV crystals occur only if the defect carries an electrical charge, either intrinsic or due to charge transfer to or from the host lattice. The absorption intensity then will be proportional to the square of the charge, since the coupling constant, in this case the dipole moment, is proportional to the charge. More details will be given in Sects. 25b and 27 d and f.

d) Localized modes, gap modes. The introduction of an impurity into a crystal will not only alter the selection rules of phonon-photon (and phononphonon) processes but also will change the phonon dynamics, i.e. the phonon frequencies and eigenvectors.

Let us take the above-mentioned U center as an example. Due to its small mass, the U center is able to vibrate at a frequency well above the bands of the pure-crystal phonons. The vibration is called a localized mode, since the frequency falls in a region in which it cannot propagate through the crystal. The rest of the crystal cannot follow the defect's rapid vibration and remains more or less static. The change in frequency and corresponding displacements of at least one phonon is thus clearly demonstrated. Analogously to the high-frequency localized mode, any vibration with its frequency outside the host-lattice frequency bands remains localized and hence is called a localized mode. Often a mode with its frequency falling in a gap between these bands is called a gap mode.

We expect an impurity mode to be the more localized the further away its frequency is from the nearest host frequency band. In KCl the upper phonon band frequency is at 217 cm -1. An example of an absorption spectrum of KCl crystals doped with D - and H - impurities is shown in Fig. 20.2. Absorption lines due to local modes are observed at 360 cm -1 and 502 cm - \ respectively, well above the host frequency bands. The structure on the wings is due to anharmonicity and will be discussed in Sect. 30j (and 30k).

If the assumption of strong localization is correct, one can calculate the isotope shift of the local-mode frequency. Suppose the defect with mass M is held in its equilibrium position by a force with force constant <1>; then its frequency will be given by

(20.1)

assuming the rest of the lattice is rigid. Since the D - center and the H- center have the same electronic configuration, the coupling to the lattice will be the


KCI :D-

>, .D

"'-~> 10

=::j 0_

E x 5 (; "" " c ::.::

"

C ::.:: 0 ~ c vi 0 .D 10·' u 0 15 c

'" 0 c E 0

(; .D

10 if> >

.D :::J <l:

5

Fig. 20.2. Local,mode absorption in KCI:H- and KCl:D-. The absorption constant is normal, ized by the UV absorption (giving the U center concentration). The peak near 500 cm -1 in the KCL: D- spectra is due to the presence of H- centers. The dotted curve represents a lorentzian fit

of the main peak at room temperature. (FRITZ, 1968)

same for both defects. We expect a frequency isotope splitting expressed by

ww'wD = (MD/MH)1/2 =V2 which compares well with 1.395 found experimentally in KCl. Indeed a ratio of

V2 to four significant figures has been observed for the U-center local mode in CsBr which lies at more than three times the maximum host frequency (OLSEN and LYNCH, 1971). In other crystals the deviation of the experimental values from V2 is generally less than 3 %; there are two main reasons for this. 1) The crystal does take part in the vibration and it does so differently for the two different defects with different frequencies. 2) Anharmonic interaction of the Ucenter mode with other phonons leads to a self-energy shift (Sects. 30g and h).

The isotope effect of the vibrational displacements can also be estimated and compared with experiment, since the infrared-absorption intensity is related to the displacements. One expects the H - local mode line intensity to be twice that of the D - line; experimentally a ratio of 1.92 is found in KCI (MIRLIN and RESHINA, 1964b; FRITZ, 1968; contrary to SCHAEFER, 1960; MITSUISHI and YOSHINAGA, 1962, who claimed a ratio of 1), which we consider as good agreement. This then may lead to the conclusion that the simple picture of the U center and the local mode probably is not totally wrong.

A list of experimentally determined local-mode frequencies due to H - and D-, as well as interstitial (see Sect. 20h) H- ions (Hn in various crystals is given in Table 20.1. A list of gap modes in alkali halides is given in Table 20.2. While the U center seems to be the only defect which causes a local mode in alkali halide crystals, a variety of defects give rise to local modes in crystals


Table 20.1. Infrared-active localized·mode frequencies (wL) and widths (2 YL) of H- and D - substitu-tional and interstitial ions in alkali and alkaline earth halides

Host Defect wL(cm- l ) 2YL(cm- l ) T(K) References

6LiF H- 1030.9 20 SIEVERS and POMPI (1967) 1020 300

LiF H- 1027 4.2 20 DOTSCH et al. (1965), 1015 18 300 DOTSCH (1969a)

D- 746 11.8 20 DOTSCH et al. (1965), 741 300 DOTSCH (1969a)

NaF H- 859.5 20 DOTSCH et al. (196..1), 858.9 <0.4 70 DOTSCH (1969a, b) 846.7 12 300

D- 615.0 1.3 20 DOTSCH et al. (1965), 614.7 90 DOTSCH (1969a) 607.5 10 300

NaCl H- <0.1 20 SCHAEFER (1960), 562.5 5.6 90 FRITZ (1965), FRITZ 565 43 300 et al. (1965)

D- 1.0 20 FRITZ et al. (1965) 408 4.0 90

300

NaBr H- 0.5 20 SCHAEFER (1960), 496, 498 17 90 FRITZ et al. (1965) 504 60 300

D- 1.0 20 FRITZ et al. (1965) 361 10 90

NaI H- 426.8 10 BAUERLE und FRITZ (1967), 430.6 100 PRYCE and WILKINSON

(MARADUDIN,1966b)

D- 317.8 10 BAUERLE und FRITZ (1967)

KF H- 725.5 100 PRYCE and WILKINSON (MARADUDIN,1966b)

KCl H- <0.1 9 SCHAEFER (1960), MITSUISHI 499; 500.0; and YOSHINAGA (1962,1963), 502 2.3; 6 90 MIRLIN and RESHINA (1964 b), 497 32; 26 300 FRITZ (1965),

FRITZ et al. (1965)

D- 0.5 20 MITSUISHI and YOSHINAGA 359, 360 2.3; 7 90; 100 (1962,1963), FRITZ (1965), 357, 360 20; 31 300 FRITZ et al. (1965)

H i- 851 20 FRITZ (1962)

KBr H- 450,446,447 6 90 MITSUISHI and YOSHINAGA 448, 444 300 (1963), MIRLIN and RESHINA

(1964b), FRITZ (1965), FRITZ et al. (1965), TrMUSK and KLEIN (1966)


Table 20.1 (continued)

Host Defect wL(cm- 1 ) 2ydcm - 1 ) T(K) References

D- 318 12 MITSUISHI and YOSHINAGA 319.7 5.9 100 (1962,1963), MIRLIN and

RESHINA(1964b), TrMUSK and KLEIN (GENZEL, 1967)

H i- 794 1.5 10 FRITZ (1962), DURR and BAUERLE (1970)

D i- 567 10 DURR and BAUERLE (1970)

KI H- 382 14.5 90 FRITZ (1965), 392 300 FRITZ et al. (1965)

D- 278.6 100 PRYCE and WILKINSON (MARADUDIN,1966b)

H i- 718 0.8 10 FRITZ (1962), BAUERLE and FRITZ (1968b)

D i- 518 10 BAUERLE and FRITZ (1968b)

RbF H- 703.1 100 PRYCE and WILKINSON (MARADUDIN,1966b)

RbC! H- 476 4.8 90 SCHAEFER (1960), FRITZ 466 300 (1965), FRITZ et al. (1965)

D- 339 3 90 FRITZ et al. (1965) 338 300

H i- 800 20 FRITZ (1962)

RbBr H- 425 8 90 FRITZ (1965), 412 300 FRITZ et al. (1965)

RbI H- 360 57 SCHAEFER (1960)

CsC! H- 425 20 DOTSCH and MITRA (1969) 424 100 417 300

D- 366 20 DOTSCH and MITRA (1969) 302 100

CsBr H- 364.08 0.4 5.5 MITRA and BRADA (1966), 366 20 DOTSCH and MITRA (1969), 364.8 5.3 63 OLSEN and LYNCH (1971) 363, 365 11 80, 100 363 35 300

D- 257.40 6.8 DOTSCH and MITRA (1969), 259.1 5.7 100 OLSEN and LYNCH (1971) 263 80 300

CsI H- 282.8 ~0.6 6.8 DOTSCH and MITRA (1969), 286.3 19 70 OLSEN and LYNCH (1971) 290 100 303 300

D- 200 20 DOTSCH and MITRA (1969) 207 ~20 100 219 300



Host Defect wL(cm- 1) 2h(cm- 1) T(K) References

CaF2 H- 965.6 <0.7 20 HAYES et al. (1965a). 957.8 8.7 290 ELLIOTT et al. (1965)

D- 694.3 2.2 20 HAYES et al. (1965a), 689.0 6.5 290 ELLIOTT et al. (1965)

Hi- 1298 300 Y ATSIV et al. (1967) Di- 934 300 YATSIV et al. (1967)

SrF2 H- 893.2 1.7 20 ELLIOTT et al. (1965) 885.3 8.3 290

D- 640.4 2.5 20 ELLIOTT et al. (1965) 635.8 5.4 290

SrCl2 H- 569 1 8 JUMEAU (1973) 571 9 77 566 200

D- 402 77

H - and La, Ce, Pr 77 See SINNEMA et al. (1974)

BaF2 H- 806.6 8.8; <0.8 20 ELLIOTT et al. (1965), HAR-798.2 14 290 RINGTON and WALKER (1971)

D- 1151.7 20 ELLIOTT et al. (1965)

BaC1F H-, D- See JUMEAU et al. (1973), SrC1F JUMEAU (1976)

LaF3 H-,D- See CeF3 JONES and SATTEN (1966) PrF3 NdF3

KCN H- See CASTRO et al. (1976)

with the wurtzite and zincblende structure; Table 20.4 based on SPITZER's (1971) compilation summarizes these data. Because many defects in the (partially) homopolar crystals have a different valency from the ions they replace additional defects must be present because of charge neutrality. Thus defectpair configurations are probable which result in rather complicated spectra and sometimes somewhat uncertain assignments. The articles by HAYES (1968); NEWMAN (1969); and SPITZER (1971) give an extensive review on local modes in semiconductors. The U-center local mode is reviewed by FRITZ (1968). A very thorough investigation of U centers in rare-earth halides has been made by ELLIOTT et al. (1965).

e) Resonant modes. We have seen above that the U center, due to its small mass, gives rise to a perturbed mode with extreme properties. At the opposite extreme, a very heavy isotopic defect is expected (BRODT and VISSCHER, 1962; KAGAN and IOSILEYSKII, 1962, 1963c; TAKENO, 1963) to give rise to a lowfrequency oscillation. Indeed, the existence of such modes has been proven by infrared-absorption measurements (WEBER, 1964; SIEYERS, 1964a, b; and many


Table 20.2. Infrared-active gap-mode frequencies WG and local-mode frequencies WL in alkali halides (modes due to molecular impurities not included)

Host Defect wG(cm- 1) T(K) References

NaI Cl- 82.9 4 CLAYMAN et al. (1969)

KI 85Rb+ 86.93 5 DE JONG (1971), DE JONG and 87Rb+ 86.41 5 VANDERELSKEN (1971) 85Rb+ 86.83-86.88 4.2 WEGDAM et al. (1973), 87Rb+ 86.17 4.2 DE JONG et al. (1973)

Cs+ 83.5 2 SIEVERS (1965a, 1969) 35Cl- 77.10 2 NOLT et al. (1967), SIEVERS 37Cl- 76.79 2 (1969), SIEVERS et al. (1965),

SHOTTS and SIEVERS (1974)

Br--Br- 73.8 2 NOLTet al. (1967), SIEVERS (1969)

81Br 88.47 4.2 WARD and CLAYMAN (1974a) 79Br 88.94 4.2

H;- 86.7 10 BAUERLE and FRITZ (1968b)

e- (F) 82.62 1.2 BAUERLE and FRITZ (1968a, b), BAUERLE and HOBNER (1970)

KBr H.-I 98.7 10 DURR and BAUERLE (1970)

e- (F) 100 10 DURR and BAUERLE (1970)

Host Defect wL(cm- 1) 2y(cm-l) T(K) References

AgCl 6Li 228.3 10 2 HATTORI et al. (1973, 1975) 7Li 218.9 10 2

AgBr 6Li 205.9 4.0 2 HATTORI et al. (1973, 1975) 7Li 191.8 4.0 2

others since) where one finds strong lines in the lower part of the phonon band. An example is given in Fig. 20.3. Again, the isotope splitting can be observed. In the example shown, isotopically pure dopants had to be used resolve the splitting. The amplitude pattern of the oscillations which give rise to these strong absorption lines does extend throughout the crystal but is very large at and near the defect (see the amplitude pattern calculated by PAGE and HELLIWELL, 1975). Hence one calls these modes quasi-localized or resonant modes (in analogy to a resonating oscillator). Only in the case of a very weak coupling can one say that the defect is sufficiently decoupled from the lattice that the lattice will not follow the vibration of the defect. For the general case one has to develop more exact criteria to predict the resonant-mode frequencies, the degree of localization, etc. This will be accomplished by the Lifshitz formalism which will be developed in Sects. 22 and 23 and which will be valid for any kind of perturbed lattice modes.

A list of experimentally observed resonant-mode frequencies in alkali halides is given in Table 20.3. In semiconductors only a few resonant modes have been observed so far, the frequencies of which are included in Table 20.4.

Sect. 20

... ~ 2.0 E u

r:: .. . (3 ........ --.. o U - .

. ~ 1.0 Q. (; .. .., ct

Types of defect and their effects

NaCI: Cu·

T = 4.2°K

-II--

Frequency (em-I)

273

Fig. 20.3. Impurity-induced absorption coefficient versus frequency for 63CU+ and 65CU+ impurities in NaCI. Curve A has 1.15 x 10'7 65CU+ ions/cm3, and curve B has 1.06 x 10'7 63CU+ ions/ cm 3 The interferometer resolution is indicated by the separation of the two arrows. (KIRBY et a!.,

1968b)

Table 20.3. Infrared-active resonant-mode frequencies OJR in alkali halides

Host Defect OJR(cm- 1) 2rR(cm- 1) T(K) References

NaCI Li+ 44.5 5 7 SIEVERS (1969), MACDONALD and KLEIN (1968), MACDONALD et al. (1969)

Cu+ 13 77 WEBER and NETTE (1966), 63CU+ 23.57 0.38* 4.2 WEBER and SIEBERT (1968), 65CU+ 23.20 0.40* 4.2 SIEVERS (1969),

KIRBYet al. (1968b)

Ag+ 52.5 10 2.3 WEBER (1964), SIEVERS 53.1 11 (1969), WEBER and SIEBERT

(1968), MACDONALD and KLEIN (1968), MACDONALD et al. (1969)

F- 59.5 <2 7 MACDONALD and KLEIN (1968), MACDONALD et al. (1969)

59.5 <2 4.2 BECKER and MARTIN (1972)

NaBr Ag+ 48.0 SIEVERS (1969,1971)

NaI Ag+ 36.7 SIEVERS (1969, 1971)

Tl+ 67; 77.4 2 SIEVERS (1964b)



Host Defect wR (cm- 1) 2rR (cm- 1) T(K) References

Cl- 5.43 ; 0.17 4.2 CLAYMAN and SIEVERS (1968b),

55 2.0 4.2 CLAYMAN et al. (1969, 1971), CLAYMAN (1971)

KCl Ag+ 38.6 7.7; 5.5 2; ? SIEVERS (1964 a, 1969)

F- 77 11 WARD and TIMUSK (1972)

KBr 6Li+ 17.71 0.60* 4.2 SIEVERS and TAKENO (1965), 7Li+ 16.07 0.43* 4.2 SIEVERS (1965b), KIRBY et al.

(1968b)

Ag+ 33.5 8.4; 4.5 2· ? , . SIEVERS (1964a, 1969)

H- 89 7 WOLLet al. (1968), DURR and BAUERLE (1970)

F- 37 2.6 11 LEVINE (see KLEIN 1968b), WARD and TIMUSK (1972)

KI Na+ 63.5 2 SIEVERS (1965 a, 1969)

Tl+ 55; 65.2; 2 SIEVERS (1964b, 1965a, 1969) 64.5

107 Ag+ 17.37 0.47* 4.2 SIEVERS (1964a), 109Ag+ 17.23 0.45* 4.2 KIRBYet al. (1968b)

Cl- ~61 2 SIEVERS (1965a, 1969), SIEVERS et al. (1965)

D- 59.4 SIEVERS (1965a, 1969)

H- 62; 61 2;7 SIEVERS (1964b, 1965a, 1969); WOLL et al. (1968)

e 68.5 4 BAUERLE and FRITZ (1968a)

RbCl Ag+ 21.4; 26.4; SIEVERS (1969,1971) 36.1

CsBr In+ 11.0 0.64 6 PRETTL and SIEP (1971 b, c)

Tl+ 16.95 1.5 6 PRETTL and SIEP (1971 b, c)

CsI In+ 12.0 0.26 6 PRETTL and SIEP (1971 b, c)

Tl+ 14.1 0.5 6 GENZEL et al. (1969), PRETTL and SIEP (1971 b, c)

SrF2 H-, D- 139.2 1.3 HARRINGTON and WEBER (1973)

BaF2 H- 285 4 15 HARRINGTON et al. (1971)

H-, D- 127.0 1.3 HARRINGTON and WEBER (1973)

MnF2 Eu+ 15.95 <0.15 2 SIEVERS et al. (1966), ALEX-ANDER and SIEVERS (1967), SIEVERS and TAKENO (1967)

* Uncorrected for experimental slit width; estimated to be too big by about 0.02 em- 1


Table 20.4. Local, gap and resonance frequencies in semiconductors (A=Absorption, R=Raman, Rf = Reflectivity, T = Tunneling, L = Luminescence)

Host Defect w(cm-l) T(K) Method References

Si 12C 611,607.5 80 A NEWMAN and WILLIS (1965), 607,604.9 300 A NEWMAN and SMITH (1969)

13C 590, 589.1 80 A 586, 586.3 300 A

14C 573, 572.8 80 A 572,570.3 300 A

Ge ~ 285 (Res.) 80, 300 R RENUCCI et al. (1971)

lOB 646.5 80 A ANGRESS et al. (1965a), TSVETOV et al. (1967), NEWMAN and SMITH (1967b,1968)

644 300 A SMITH and ANGRESS (1963), AN-GRESS et al. (1964, 1965a), SPITZER and WALDNER (1965), BALKANSKI and NAZAREWICZ (1966), BALKANSKI et al. (1968 b)

642 300 R NAZAREWICZ et al. (1971), BESERMAN et al. (1972)

11B 622.8 80 A Same as for lOB 620.2 300 A 619 300 R

lOB_6 Li 536, 537, 587, 683 80 A BALKANSKI and NAZAREWICZ (1966), TSETOV et al. (1968)

534, 584, 681 300 A BALKANSKI and NAZAREWICZ (1964,1966), SPITZER and WALDNER (1965), BALKANSKI et al. (1968b), DEVINE and NEWMAN (1969)

lOB_7Li 523, 525, 587, 683 80 A Same as for lOB_6 Li 522, 584, 681 300 A

IlB_6 Li 536, 537, 566, 656 80 A CHRENKO et al. (1965), BALKANSKI and NAZAREWICZ (1966), TSVETOV et al. (1968)

534, 564, 653 300 A BALKANSKI and NAZAREWICZ (1964,1966), SPITZER and WALDNER (1965), ANGRESS et al. (1965 a), BALKANSKI et al. (1968b), DEVINE and NEWMAN (1969)

IlB_7Li 523, 525, 566, 656 80 A Same as for 11 B-6 Li 522, 564, 655 300 A

lOB_lOB 570.0 80 A NEWMAN and SMITH (1967 a, b, lOB_11B 560.0 80 A 1968) 11B_11B 552.5 80 A

lOB_P 622/3, 653/5 80 A NEWMAN and SMITH (1967b, 1968), 11B_P 600,629/31 80 A TSVETOV et al. (1967, 1968)

lOB_As 625,662 80 A NEWMAN and SMITH (1967b, 1968)



Host Defect w(cm- 1) T(K) Method References

llB-As 604,627,637 80 A NEWMAN and SMITH (1967b, 1968), TSvETOvet al. (1968)

l°B-Sb 635,668 80 A Same as for B-As llB-Sb 612,643 80 A

AI-6Li 525 (Res.) 300 A DEVINE and NEWMAN (1969) AJ-1Li 520 (Res.) 300 A

Ga-6Li 521 (Res.) 300 A DEVINE and NEWMAN (1969) Ga-7Li 515 (Res.) 300 A

AI-AI 469,431 300 A DEVINE and NEWMAN (1969) Al 242 300 A

P 154,420,484 (Res.) 4 T SCHEIN and COMPTON (1970) 441,491 (Res.) 80 A ANGRESS et al. (1968)

As 366 (Res.) 80 A ANGRESS et al. (1968)

O,C-O, see NEWMAN (1969), O-Li BOSOMWORTH et al. (1970)

SiC N 635 4 T,R SCHEIN and COMPTON (1970)

Ge Si ~389 77 R RENUcci et al. (1971)

~389 300 R FELDMAN et al. (1966), RENUCCI et al. (1971)

389 300 A SPITZER (1971)

Si-Si 476,448 300, 77 R FELDMAN et al. (1966), RENUCCI et al. (1971)

lOB 571 200 A NAZAREWICZ and JURKOWSKI llB 547 200 A (1969)

l°B-Li 518.5,610 200 A NAZAREWICZ and JURKOWSKI llB-Li 497,582.5 200 A (1969)

Ga-6Li 379,405 80 A COSAND and SPITZER (1967) Ga-7Li 356,380 80 A

Ga-P 350 80 A COSAND and SPITZER (1967)

P 343 80 A COSAND and SPITZER (1967)

160 862 80 A WHAN (1965)

855 300 A KAISER (1962)

180 818 80 A WHAN (1965)

AlAs Ga 252 (Res.) 300 Rf ILEGEMS and PEARSON (1970)

GaP 12C 606 20 A HAYEset al. (1970) 605.7 77 A THOMPSON and NEWMAN (1971)

14C 564 20 A HAYES et al. (1970)

28Si(Ga) 453; 464.9, 80 A SPITZER et al. (1969 a) 465.6*



Host Defect w(cm-1) T(K) Method References

29Si(Ga) 461.1 * 80 A THOMPSON and NEWMAN (1971) 3OSi(Ga) 456.6* 80 A (* heavily doped samples)

B 284 80 A GLEDHILL et al. (1981)

SieGal 253 80 A GLEDHILL et al. (1981)

lOB 593.6 20 A HAYES et al. (1969) 293.8 (Gap) 20 A

593.3; 592; 592.7 80 A HAYES et al. (1969), SPITZER et al. (1969a),

293.5 (Gap) 80 A THOMPSON and NEWMAN (1971), MORRISON et al. (1974)

593.8 100 R HON et al. (1970)

594 100 A ZAKRZHEVSKII et al. (1971) 590 300 A

liB 571.0 20 A Same as for 11 B 284.2 (Gap) 20 A 570.7; 569; 570.0 80 A 283.8 (Gap) 80 A 569.7 100 R 571 100 A 568 300 A

B 284 80 A GLEDHILL et al. (1981

Al 443.3 80 A SPITZER et al. (1969a), THOMPSON and NEWMAN (1971)

444.7 100 R HON et al. (1970)

14N 491.6 77 A NEWMAN and THOMPSON (1971) 14N ~492 1.6 L THOMAS and HOPFIELD (1966) 15N ~476 1.6 L

As 270 (Gap) 20 A HAYES et al. (1969, 1970) 273 80 A GLEDHILL et al. (1981) 272 300 Rf VERLEUR and BARKER (1966),

BARKER (1968) ~270 300 R STRAHM and WHORTER (1969)

267, 275, 373, 381 300 R HAANSTRA and VINK (1973)

0 464 80 A SPITZER et al. (1969 a), ZAKRZHEVSKII et al. (1971)

464.5 100 R HON et al. (1970)

Mn 320 300 R HAANSTRA and VINK (1973)

Be 527 90 A BERNDT et al. (1975)

GaAs SiOa 384 80 A LORIMOR and SPITZER (1966), SiAs 399 80 A SPITZER and ALLRED (1968 a, b),

SPITZER et al. (1969 b)

SiOa-SiAs 367,393,464 80 A SPITZER and ALLRED (1968 b), SPITZER et al. (1969b)

SiOa-6Lioa 374, 379, 405, 470, 80 A LORIMOR and SPITZER (1966), 480,487 SPITZER and ALLRED (1968b)

SiOa-7Lioa 374, 379, 405, 438, 80 A 448,455



Host Defect w(cm- l) T(K) Method References

SiGa-CuGa 374, 376, 399 80 A LORIMOR and SPITZER (1966), SPITZER and ALLRED (1968 b)

SiGa-ZnGa 378, 382, 395 80 A ALLRED et al. (1968), SPITZER (1971 )

12C 582 80 A BROZEL et al. (1978) 13C 561 80 A

lOB 540.2 77 A THOMPSON and NEWMAN (1972), lIB 517.0 77 A MORRISON et al. (1974) B 123 300 A ANGRESS et al. (1980)

BGa-BAs see THOMPSON and NEWMAN (1972)

Al 362 80 A SMITH et al. (1966), LORIMOR et al. (1966), SPITZER (1967), LORIMOR and SPITZER (1967a, b), SPITZER and ALLRED (1968b)

359 300 A LORIMOR et al. (1966)

356 300 Rf ILEGEMS and PEARSON (1970)

P 355.4 80 A LORIMOR et al. (1966), KLEINMAN et al. (1966), SMITH et al. (1966), SPITZER (1967)

357 80, 300 R USHIODA (1970)

359, 353 300 A LORIMOR et al. (1966), CHEN et al. (1966)

352 300 Rf VERLEUR and BARKER (1966), CHENet al. (1966)

Zn-6Li 361,386/5,406/4, 80 A HAYES (1965), 432/3 LORIMOR and SPITZER (1967b)

Zn-7Li 340, 361, 378, 405 80 A

Cd-6Li 378/7,408/1,424/3 80 A HAYES (1965), LORIMOR and SPITZER (1967b)

Cd-7Li 354, 375, 395 80 A LORIMOR and SPITZER (1967b)

0-6Li 402 80 A HAYES (1965) 0-7Li 395 80 A

Te-6Li 419,510 80 A HAYES (1965), Te-7Li 391,475 80 A LORIMOR and SPITZER (1967a)

Mg-6Li 344,391,423 80 A L. SKOLNIK (SPITZER 1971) Mg_7Li 344, 364, 390 80 A

Mn-6Li 391,413,419 80 A LORIMOR and SPITZER (1967b) Mn-7Li 365, 386, 391 80 A

6Li 489, 406, 409.5, 80 A HAYES (1965), LEVYet al. (1968) 426,452

7Li 364, 379, 383, 398, 80 A 421



Host Defect m(cm- l) T(K) Method References

GaSb Al 316.7 80 A HAYES (1964) 313.8 300 A

In 199 (Res.?) 300 Rf BRODSKYet al. (1970)

P 324.0 80 A HAYES (1964) 321.4 300 A

As 240 (Res.?) 300 Rf LUCOVSKY and CHEN (1970)

InP lOB 543.5 80 A NEWMAN et al. (1970) llB 522.8 80 A

As 223 (Gap?) 300 Rf KEKELIDZE et al. (1972)

InAs Ga 240 300 Rf BRODSKY and LUCOVSKY (1968), LUCOVSKY and CHEN (1970)

P ~293 30 Rf OSWALD (1959) 203 300 Rf

InSb Al 295.7 80 A GOODWIN and SMITH (1965), 292.8 290 A SMITH et al. (1966)

P 293 80 A L. SKOLNIK (SPITZER 1971)

As 200 300 Rf LUCOVSKY and CHEN (1970)

ZnS Al 438 80 A MITSUISHI et al. (1970)

Se ~220 (Gap) 100 Rf,A,R BRAFMAN et al. (1967, 1968)

Be 489.8 100 A,R MITSUISHI et al. (1970), MANABE et al. (1971)

ZnSe 6Li 412 100 A IBUKI et al. (1967), 7Li 393 100 A MITSUISHI et al. (1970)

Al 359 100 A IBUKI et al. (1967), MITSUISHI et al. (1970), see also DUTT und SPITZER (1976)

S 297-298 4.2, Rf,A,R BRAFMAN et al. (1967, 1968), 80, 300 BALKANSKI and BESERMAN (1968),

BESERMAN (1969), BESERMAN and BALKANSKI (1970a)

Te 182 (Res.) 300 Rf BESERMAN (1969), BESERMAN and BALKANSKI (1970a)

Mn 199 (Res.) 300 Rf BESERMAN (1969), BESERMAN and BALKANSKI (1970a)

Be 449.7 100 A,R YUASA et al. (1970), MITSUISHI 447.2 300 A,R et al. (1970), MANABE et al. (1971)

24Mg 352 77 A MITSUISHI et al. (1970) 25Mg 345 77 A 26Mg 334 77 A

ZnTe Be 415.0 100 A,R MANABE et al. (1971)

S 269-270 300 Rf BESERMAN (1969), BESERMAN and BALKANSKI (1970a)



Host Defect w(cm- 1) T(K) Method References

Al 313 80 A MITSUISHI et al. (1970)

Cd ~148 300 Rf VODOPYANOVet al. (1972)

CdS Se 182.5, 187 (Gap) 300 Rf BALKANSKI and BESERMAN (1968), BESERMAN and BALKANSKI (1970 b)

182(1f), 187(.1) 4.2 Rf BESERMAN (1969)

~190 80 R CHANG et al. (1969) ~188 80, 300 Rf,R PARRISH et al. (1967)

Be 446.1 (11),452.7(.1) 100 A,R MANABE et al. (1973)

6Li 474 100 A MITSUISHI et al. (1966, 1970) 7Li 457 100 A

Mn 297 (Gap?) 300 R BESERMAN et al. (1971)

CdSe S 270,269.5 15 Rf VERLEUR and BARKER (1967)

260,270 25 A BALKANSKI et al. (1968a)

~270 80 R CHANG et al. (1969), PARRISH et al. (1967)

~ 266.5 (If), 4.2 Rf BALKANSKI et al. (1968a), 269 (.1); 268 BESERMAN (1969), BESERMAN and

BALKANSKI (1970b), PARRISH et al. (1967)

~268 300 R PARRISH et al. (1967)

Be 441.2(.1),419.9(11) 100 A,R MANABE et al. (1973)

Li 385 4.2 Rf BESERMAN (1969)

CdTe Se 162 4.2 Rf BESERMAN (1969)

160,170 25 A BALKANSKI et al. (1968a)

170 300 A BESERMAN and BALKANSKI (1970a)

Li? 270 25 A BALKANSKIet al. (1968 a)

Be 391; 61 (Res.) 4 A HAYES (1968), HAYES and SPRAY (1969), SENNETT et al. (1969), cf. TALWAR and AGRAWAL (1974)

Mg 266.3 D.T.F. MARPLE (GAUR et al. 1971)

AI 299 80 A DUTT et al. (1976)

S 250 300 A ARTAMONOV et al. (1976)

P 269,308,322 80 A DUTT and SPITZER (1977 a, b)

Zn 166 300 Rf V ODOPY ANOV (1972)

f) Off-center and molecular defects: Tunneling motion. There is a number of substitutional defects which do not occupy the exact place of the replaced host crystal ion; instead, their center is slightly shifted off in one of the main symmetry directions. For example, the smallness of the Li+ defect causes a potential weakening which is so strong in some alkali halides that the equilib-


rium position of the Li + ion is slightly shifted off the corresponding regular lattice site, as is schematically shown in Fig. 20.1c. The point symmetry of the crystal is thus reduced. For reasons of symmetry there are generally more than one of the equivalent "off-center" potential minima, between which the defect may undergo quantum mechanical tunneling.

Similarly, molecular impurities, see Fig.20.1d - like an OR - molecule substituting for a halide ion in an alkali halide - may have different equivalent equilibrium orientations and therefore behave in some respects like an off-center defect.

Rather interesting effects are connected with the tunneling properties of molecular and off-center defects. For example, the equilibrium sites of the substitutional Li + ion in KCI are shifted off the ideal lattice sites in the eight (111) directions, which are shown in Fig. 20.4. The Li+ ion can occupy any of these sites and is able to tunnel from one site to another. It has been found that the tunneling occurs most probably from one site to one of its three adjacent ones, other transitions being less probable. The tunnel transitions split the eightfold tunneling ground state of the KCI:Li+ system. Analogous effects occur for other off-center and molecular systems.

Transitions between the tunneling states of molecular or off-center defects generally occur at very low frequencies. They were first discovered indirectly in thermal conductivity measurements by SEWARD in 1965. KIRBY et al. (1970) observed the tunneling transition of the off-center Li+ in KCI directly in infrared-absorption experiments. The small tunnel splitting of a few wavenumbers (or less) can be increased by the application of stress or electric fields; the position of the absorption line, therefore, shifts by an amount proportional to the field (in the region where the field-induced splitting is much larger than the tunnel splitting); this is shown in Figs. 20.5 and 20.6. More recently, the small frequency of the tunneling motion of CN - and OR - molecules and its stress and electric-field dependence has been observed to large accuracy as "para-

rLi+= 0.60A

Fig. 20.4. The local atomic arrangement for the system KCl:Li. The eight potential wells lie at the comers of a cube of 1.4 A edge length. The Li + tunnels predominantly along the dashed cube edges. The Cl ions at the centers of the large cube faces have been omitted for clarity. p indicates

the electric dipole moment. (NARA YANAMURTI and POHL, 1970)

282

! .4 c

" 'u ~ .3 8 c ~ .2 Q.

! "

Lattices with point defects

•• -15.8 kV fem

Cb-30.6kV/cm 00

KCI : Li' CI I. 2"K Eoe II (IOOJ EIRII Eoe ~f-

:.-51.5kV/cm

noo71.4 kV/cm

0 " "" V o " ,

"'I/' 0""",,; ....... ""'/

, ""-'

Z 3 4 5 6 7 8 9 Frequency (em-I)

Sect. 20

Fig.20.5. KCI: 6Li tunneling absorption line for several values of the external [100J electric field. (KIRBY et aI., 1970)

c:: .~

14r---~----~---'-----r----r---~--~

12

6

KCI.Li 6 CI

Eoc II [100]

EIR II Eoc Nlj+ : 4.6 xlO l7/cm 3

Q.

~ 4 .Q

<t

°0~--_2iO----4~0~--~60~--~8~0----1~00~--1~2~0--~

E(kV/cm)

Fig. 20.6. [100J field dependence of a KCI: 6 Li tunneling absorption line. These data include points at both 1.2 and 4.2 K. The solid line is calculated from the model of GOMEZ et al. (1967).

(KIRBY et aI., 1970)

elastic" sidebands of the absorption line due to internal molecular vibrations (LOTY, 1974a, b; BEYELER, 1975) and directly by phonon scattering techniques using phonons from tunable superconducting tunnel junctions (WINDHElM and KINDER, 1975; WINDHEIM, 1976).

Recent interest in off-center defects has arisen in the context of ferroelectric phase transitions. HALPERIN and VARMA (1976) suggesting that one should expect an influence of defects on the critical phenomena caused a number of


investigations (HOCK et aI., 1978, 1979; HOCK and THOMAS, 1977, 1979; SCHMIDT and SCHWABL, 1976, 1978 a, b), in particular they argued that the socalled central peak in scattering spectra should reflect the relaxation behaviour of off-center defects. Indeed, some behaviour of (nearly) ferroelectric materials can (and possibly must) be understood in terms of elaxing defects (HOCHLI et aI., 1978, 1979; YACOBY and JUST, 1974; YACOBY, 1978a, b, 1981). It seems possible that cooperative behaviour of these defects occurs.

The tunneling and rotational aspects of molecular and off-center defects are outside the scope of lattice dynamics and their implication for absorption spectra. They are covered by some detailed review articles (LOTY, 1967; NARAYANAMURTI and POHL, 1970; BRIDGES, 1975; PRESS, 1981), so we do not enter this field any further in this article.

g) Internal vibrations of molecular defects. In addition to the tunneling motion, the molecular impurities have internal vibrations as additional degrees of freedom. Since the free molecules become distorted and coupled to the surrounding ions when built into the lattice, the frequencies are slightly changed from the free-ion frequencies (Sect. 26b). The rotational degrees of freedom of a free molecule are changed to either hindered rotations or librations, depending on how strongly it is coupled to the lattice (DEVONSHIRE, 1936; SAUER, 1966; HOLLER and KROLL, 1975).

The internal vibrations of e.g. the OH - substitutional molecule in alkali halides have been observed by infrared as well as Raman techniques. (The stretching vibration of an O2 molecule would be Raman active only.) The OHstretching vibration is found at about 3600 cm -1, the libration is found in the 200 to 400 cm -1 region. Some of the observed frequencies are listed in Table 20.5. Unfortunately, the stretching vibration has not been measured for a free OH - ion; therefore, the frequency enhancement due to coupling of the molecule to the lattice and hence the coupling itself cannot be estimated. The stretching vibration of the neutral free OH ion is at 3569 cm -1 (HERZBERG, 1950). All predicted values (see KLEIN et aI., 1969) for the stretching vibration of the free ionic molecule are above the value of 3468 cm -1 as recently found by STERK and HANSON (1971) for AgCl:OH-. This value should, according to the considerations in Sect. 26, place an upper limit to the free-ion vibration. In addition, anharmonic effects shift the mode frequencies. The OH- and ODvibrations in SrTi03 at 3495cm- 1 and 2582cm- 1 at room temperature (e.g. HESSE and KApPHAN, 1978) are split into three components below the transition temperature (KLUKHUHN et aI., 1970; KApPHAN et aI., 1980). The OHvibrations in hydrogenated MgO are much lower (about 3300 cm -1), and the proton is supposed to sit in <111) off-center positions (see FREUND, 1976). In the same frequency region the vibration of OH- in Ti02 is seen, and the isotope effect due to replacement of the hydrogen by deuterium and tritium has been observed by BATES and PERKINS (1977).

Anharmonic coupling of the internal molecular vibrations to the lattice modes causes sidebands to appear on the wings of localized vibrational lines due to molecules and other defects (see Sect. 30j). METSELAAR and V AN DER ELSKEN (1966, 1968) remarked on the information contained in these sideband spectra.


Table 20.5. Stretching (ws) and librational (wL ) frequencies of the OH- molecule substituted in various alkali halides as observed by infrared (IR) and Raman techniques. Listed is also the separation of the librational sideband frequency (wL +S) from the stretching line frequency. Data are taken

at liquid He (LHT) and room temperature (RT)

Host Ws IR Ws Raman

LHT RT

LiF 3731 a, b

NaF 3728' NaCl 3654.5 d 3650-3660 d

NaBr 3626 d

KCl 3641.0 d 3639 d , ,

KBr 3617.5 d 3611 d, ,

KI 3603 d

RbCl 3632.5 d

CsBr AgCl 3468.3 f

a STOEBE (1967)

LHT

3651.6" j

3643" "

3618'

b GUCKELSBERGER and NEUMAIR (1970) , MEISTRICH (1968) d WEDDING and KLEIN (1966, 1969) , FENNER and KLEIN (1969) f STERK and HANSON (1971) g CALLENDER and PERSHAN (1969,1970)

RT

3638'

3631' 3628" 3613' 3598"

h WEDDING and KLEIN (1966), KLEIN et al. (1969) , PRETTL and SIEP (1970) j PEASCOE and KLEIN (1973)

LHT RT LHT RT LHT RT

390.0 h

297.5 h 305' ~300"

312.7" 309.7 h 313'

284 h 274h 272.0h 270.5 h

289' 221.0 f

CUNDILL and SHERMAN (1968; see also references therein), for example, deduce from their sideband spectra that the alkali halides KBr, KI, RbBr, and RbI which have a phonon gap, do not have a gap in their high-pressure CsCl phase.

Coupling of the OH - molecule to the lattice causes the stretching vibration to vary between roughly 3450 and 3750 cm -1 in different lattices. The librational frequency, which is entirely due to that coupling, varies between 220 and 390 cm -1. Librationallines have been observed by infrared (HARRISON and LUTY, 1967; KLEIN et aI., 1969; PRE TTL and SIEP, 1970) and Raman (FENNER and KLEIN, 1969) techniques. Anharmonic coupling between libration (w=wd and stretching vibration (w=ws) causes an absorption at WL+S~WL +ws (CHAU et aI., 1966; KLEIN et aI., 1969; STERK and HANSON, 1971). The libration of the OH - ion in KBr: OH - seems to be very anharmonic because its position and width vary strongly with temperature. In Fig. 20.7 the direct librational absorption line at two different temperatures is shown. Figure 20.8 shows the same band as a sideband of the stretching vibrational line at 3617.5 cm -1. This sideband exhibits the same behaviour upon temperature variation. There is another structure at about 30 cm -1 separation from the stretching line which in most crystals can be seen directly in the far-infrared region as the so-called "Xline" or "30 wavenumber line" or "non-Devonshire line" (BOSOMWORTH, 1967; see CHIBA and AHA RATA, 1979 for one interpretation). In addition, a

Sect. 20

1- 5

~ ~4 8 z 3 o ;: D-o: 2 ~ CD

-<I

o


KBr:OH-

240 ppm 0-0 aOK

_70'K

285

340

Fig. 20.7. The direct OH- librational absorption band in KBr at two temperatures. (KLEIN et aI., 1969)

>f-U5 Z W o -l .5 « ~ f-a... o

KBr:OH 600 ppm 5°K_

78'K .............

-I em

Fig. 20.8. The infrared-absorption spectrum of KBr containing 600 ppm of KOH. Crystal thickness, 16 mm. (WEDDING and KLEIN, 1969)

stretching-vibration overtone has been observed in KBr: OR- (WEDDING and KLEIN, 1969). The isotope shifts of the librational and stretching vibrational band are found to be in good agreement with the predicted ratio of 1.375 which is the square root of the moments of inertia and reduced mass.

Absorption lines have been observed in the frequency-gap region in KI:OR- (RENK, 1965, 1966, 1967; SIEVERS and LYTLE, 1965; GRISAR et aI., 1967; BECKER, 1970), and various lines have been found in NaCl: OR - in the very far-infrared frequency region (KIRBY et aI., 1968a).

A review of most of the vibrational and librational properties together with the properties which are based upon reorientation kinetics (tunneling) has been given by LUTY (1967).

While the OR molecule seems to have been investigated very thoroughly, data on other systems are far less complete. The CN - ion has much in common


with OH- (CALLENDER and PERSHAN, 1969, 1970), but few data on vibrational properties exist (see also CHI and NIXON, 1972; MAURING and REBANE, 1975; TSYASHCHENKO et a1., 1976; for SH- and SD-; RADHAKRISHNA et a1., 1976, for VO+ +). Rather more data exist for the N0Z- and N03 ion in alkali halides. Some recent references are CALLENDER and PERSHAN (1969, 1970); EVANS and FITCHEN (1970); DEMYANENKO and TSYASHCHENKO (1969); MAKSIMOVA (1969); EIJNTHOVEN and VAN DER ELSKEN (1969); BRUINING and VAN DER ELSKEN (1975); REBANE (1975); REBANE et a1. (1975, 1977); HALDRE et a1. (1975); BYSTROV et a1. (1976); JODL and HOLZAPFEL (1978); MAKSIOMOVA and RESHETNYAK (1978). More complex molecules like SO;(RAbHAKRISHNAN, 1971; MAKSIMOVA and RESHETNYAK, 1971; TSYASHCHENKO et a1., 1973; KRASNYANSKII, 1977; BOIKO et a1., 1977a,b); MnO; (MANZELLI and TADDEI, 1969; RADHAKRISHNA and HARIDAsAN, 1979); CrO;-, Se03 -, and SeO;- (DEMYANENKO et a1., 1970, 1971); and CNO(DEcms et a1., 1965; KONDILENKO et a1., 1970, 1975; SCHETTINO and HISATSUNE, 1970; SMITH et a1., 1973; TSYASHCHENKO et a1., 1973, 1975a, b; ERMOLENKO et a1., 1976) exhibit a larger number of lines than just the two lines, the stretching and (the twofold degenerate) librational mode, as observed in the OH - doped systems. In particular, phonon sidebands on the wings of the internal modes are seen in many cases and analyzed occasionally (for example ROLFE et a1., 1973). The authors apologize for making most incomplete reference to these experimental data.

Theoretical considerations are deferred to Sect. 26.

h) Interstitials. Defects which are added to the crystal and whose equilibrium positions are located between regular lattice sites, as in Fig. 20.1 e, are called interstitials. Since the number of particles is increased as in the previous case, interstitials are somehow similar to molecular defects. This will become clearer when we treat their vibrational properties in Sect. 26.

Local and gap modes due to hydride (and later also deuteride) ions at interstitial sites in alkali halides were first observed by FRITZ (1962). The interstitial hydrogen (Hn ion in KBr causes an infrared-active local mode at wL

=794 cm- 1 and a gap mode at wG=98.7 cm- 1 . Anharmonic coupling between these modes causes absorption lines at a frequency of about wL + wG' At low temperatures the line at wL + wG can be observed (see Sect. 30j) as a sideband with a separation of 102 cm -1 from the H j- induced local mode at 794 cm - \ see Fig. 20.9. Anharmonic effects must be responsible for the difference between 102 and 98.7 cm -1. The point symmetry here is 7~, which allows modes of r lS

symmetry to be both infrared- and Raman-active (DDRR and BAUERLE, 1970). This is in contrast to GROSS and BRON (1967) who concluded that the H j- ion is shifted off the Td center position. In KI with H j- ions, BAUERLE and FRITZ (1968b) have seen analogous lines at wG =86.7 cm- 1 in the far-infrared region and as a sideband with a separation of 89 cm -1 from the local-mode line (WL

=719 cm- 1). Local-mode frequencies of H j- ions in various alkali halides are included in Table 20.1.

The interstitial oxygen atom in silicon crystals has been found to be slightly off the line connecting: two neig:hboring: atoms. The position of the oxygen

III

Sect. 20

-50

0.8 r- I

Far lR KBr

0.6r- _ ) Hi (U1 - centers T =9°K

0.4r-

0.2f-

o


Wavenumber ii '(em-')

50 I

100 I

gapr~

II resolution

~ 0~-------------4----------------------~~~~--~ .~ Near IR a. KBr o

0.8 r- Hi (U1 ) - centers vL = 794 em-' •

0.6r- T = 9°K

0.4r-

0.2 r-

gap r'-I

II resolution

o -50 j \.~

o 50 100 Wavenumber ii-iiL (cm-')

287

150 I

115 modes

@ , , I

H- ,

//6---- ---

/i / w '"" I ....

~-~-- ---,/ ! ~

150

Fig. 20.9. Far-infrared and high-frequency local-mode absorption due to Hi(U 1) centers in KBr. (DURR and BAUERLE, 1970)

".--, I \ \ J , /

Fig.20.10. Schematic representation of interstitial configuration of oxygen (filled circle partly visible) in silicon. The oxygen is in the plane of the paper. Silicon atoms above the plane of the paper are represented by full circles and below the plane of the paper by dashed circles. A < 111 >

crystal axis is perpendicular to the plane of the paper. (BosoMwoRTH et aI., 1970)

relative to its nearest neighbors is similar to that in the H 2 0 molecule. Figure 20.10 shows the local arrangement of atoms. A number of lines are seen in the infrared frequency region. In particular, one observes absorption lines due to the three normal modes of the Si - 0- Si "molecule", although the assignment by various authors has been contradictory. A thorough investigation and review of


the oxygen interstitial properties, like far- and near-infrared absorption and effects due to isotopes and pressure, are given by BOSOMWORTH et aI. (1970).

i) Effects of defect clusters and defect concentration. As the concentration of defects in a lattice is raised, the probability of formation of defect clusters is increased, either because of considerable interaction between defects or because of increasing statistical probability. In the simplest case defect pairs are formed. These may be either two defects of the same kind or of two different species intentionally introduced into the crystal. Similarly, larger clusters of three or more defects may be formed. In any case, a defect pair (or triplet, etc.), firstly, has a local symmetry different from that of a single defect and, secondly, has different vibrational properties. If a single defect, for example, has a localized vibration, the neighboring defect changes the lattice response to this mode such that its frequency is shifted. In other words, a pair mode has a different frequency than a corresponding single defect mode. A satellite of the local-mode absorption peak is thus created. The further away the second defect is from the first, the less is the splitting (TAKENO, 1965). If the dynamical interaction between a defect and most of its surrounding defects is sufficiently weak, the corresponding lines are shifted by amounts smaller than the width of the single-defect line. Their main effect is then to contribute to the width of the line. Localized, gap and resonance vibrations due to pairs of defects have been observed in H- - and D- -doped alkali halides (DE SOUZA et aI., 1970; DE SOUZA and LOTY, 1973; ROBERT and DE SOUZA, 1974; ROBERT et aI., 1975; SCHNEIDER, 1974), in NaCI:Ag+ (MOLLER et aI., 1970), KCI:Na+ (TEMPLETON and CLAYMAN, 1971, 1972), NaCI:F- (BECKER and MARTIN, 1972) NaCI:KF (ISHIGAMA et aI., 1972), and KI:Na+, Cl-, Be (WARD et aI., 1974; WARD and CLAYMAN, 1974a). There is some controversy of a possible defect-pair resonance observed in KI: Rb+ (WARD and CLAYMAN, 1974c) which is also interpreted as due to a maximum in the density of states (DE JONG et aI., 1973; KAPER and VANDER ELSKEN, 1975).

Theoretical work on pair vibrations has been done using Green function techniques (MONTROLL and POTTS, 1956; TAKENO, 1962b, 1965; MART1N, 1967b; ELLIOTT and PFEUTY, 1967; PFEUTY, 1968; GRIMM et aI., 1972; HARIDASAN et aI., 1973; STRIEFFLER and JASWAL, 1969; BEHERA and PATNAIK, 1975; PATNAIK and BEHERA, 1976; GUPTA and MATHUR, 1976; and others) and the molecular model (JASWAL, 1972; WARD and CLAYMAN, 1974b, c; TEMPLETON and CLAYMAN, 1976; and others).

Charged defects in homopolar crystals usually have to be accompanied by oppositely charged defects for reasons of charge compensation. In fact, much of the experimental information is related to vibrations of pairs of defects. This field is extensively reviewed by NEWMAN (1969). Some recent results are obtained for Ge by BROZEL et aI. (1975, 1977); and theoretically by AGRAWAL and TALWAR (1978) and AGRAWAL et aI. (1979); for GaAs by BROZEL et aI. (1978); and for ZnS by KROL et aI. (1978a, b). The group-IV semiconductors are necessarily doped with different kinds of defects; in other crystals, especially in the alkali and alkaline-earth halides containing U centers, additional impurities are deliberately added. Positioned at different neighbor positions, these additional defects cause

Sect. 21 Information contained in defect-induced spectra 289

various local-mode lines to split off the pure V-center local-mode line (see the references in Sect. 29 c).

The effects of many defects, even though of small concentrations, on the dynamical properties will be the subject of Sect. 28. If the concentration of defects can be varied over most of the values between zero and unity, the corresponding systems are called mixed crystals; these will be the subject of Sect. 29.

As the dynamical properties are changed as a function of defect concentration, so are the static properties altered as well. In Sect. 31 some results for elastic constants and for the thermal expansion in the limit of low defect concentrations will be presented.

j) Dislocations, surfaces. While the types of defects mentioned above are quite localized in the crystal lattice, there are imperfections which extend over a wide range: dislocation lines and planes. Figure 20.1f shows a dislocation point in a two-dimensional lattice. Because of their non-localization dislocations are mathematically not as easy to treat as the defects mentioned above.

Since in numerical calculations the crystal often is thought of as having periodic boundary conditions, a crystal surface represents a perturbative plane, cf. Fig. 20.1 g. As a matter of fact, thin films and small crystals are either looked at as highly (but regularly) perturbed crystals or as large molecules with only partial reference to an unperturbed crysta1. This field is reviewed by LIFSHITZ and KOSEVICH (1966) and RUPPIN and ENGLMAN (1970) in the two different ways of interpretation and will be taken up in Sect. 32.

21. Information contained in defect-induced spectra. This section will be a short summary of the utility of defect-induced spectra. Since the mathematical details will be presented later, the conclusions of this section are preliminary in character and will be supplemented by the considerations of Sects. 25 and 27. On the other hand, in this section we try to give a motivation for the following sections.

The study of defect-induced spectra gives information on the properties of both defect and host. Special defect properties like light mass and/or strong coupling of the defect to the lattice give rise to localized modes which can absorb infrared radiation. The frequency of these modes, therefore, allows conclusions about the defect-host coupling strength. For example, the localizedmode frequency 0) due to V centers in alkali halides is calculated from (20.1) to be too high by a factor of about 1.4, assuming host-lattice coupling constants <1>

between defect and host lattice. From (20.1), <1>=0)2 M, one then concludes that the actual coupling is reduced by a factor of about 2 (see Sect. 22a for more details).

Other vibrations are less localized and involve motion of the lattice as well as the defect. Since there is a comparatively small number of defects, these vibrations are characteristic for the host lattice, whose properties are then reflected in an infrared or Raman spectrum. If one neglects the frequency dependence of the transition matrix elements, the spectra should reflect the density of states with a series of critical points (LAX and BURSTEIN, 1955). In fact, this is very clearly seen in the far-infrared absorption spectrum induced by the natural abundance of isotopes. KLEIN and MACDONALD (1968) have


measured the absorption in 6Lix7Li1 _ xF and Na35CI/7Cll_x crystals where the defects are purely isotopic. See POWELL and NIELSEN (1975) for h.c.p. Hl_xDx crystals. Similar effects occur for non-isotopic defects (see references on page 292, 293).

The infrared spectrum of 6Lix 7Li1 _ xF is shown in Fig. 21.1. A critical point at 252.2 cm -1 is clearly seen and has been tentatively assigned to the TA(X) phonon. (In the breathing-shell model the TA(X) phonon has a frequency of 238 cm -1.) Since the defect-induced absorption is approximately proportional to the concentration of defects, the spectrum is weak for x ~ 0 arid for x ~ 1, the intensity increases with x approaching the value !. The variation of the absorption spectrum with concentration x will be the subject of Sect. 28.

A theoretical study of the critical points in Na35CV7Cll_x as observed in the infrared-absorption spectrum shown in Fig. 21.2 has been made by MAcDONALD and KLEIN (1968) and MACDONALD et al. (1969). They found that the peaks in the absorption spectrum (dots in Fig. 21.2) could not be reproduced in a calculation (broken curve in Fig. 21.2) using phonons of a shell model in the form generally used to fit the neutron-spectroscopy data. In order to reproduce the experimental curve (in fact, quite well), Macdonald et al. had to reduce the elastic constants beyond the experimental error limits in order to obtain good agreement (full curve in Fig. 21.2). In so doing, the dispersion curves are indeed slightly pushed towards the actual dispersion curves which were measured by RAUNIO et al. (1969) after the calculations of Macdonald et al. Otherwise the model of Macdonald et al. has the deficiency of reproducing neither the elastic

20.0

10.0

'E 2 5.0 -..... Z ;:! en ~2.0 u Z Q 1.0 ..... Il.. 0:

~0.5-aJ «

0.2 ....

ANGULAR FREQUENCY Ud3rad:(sec) 3.0 4.0 5.0

. "

I

" . L " • -11 :e.-e.

(/~v.:"/ ... ~ -ir .". .4 +

/ " : . • ••• +)(

.:-/ ~ "'xx

•• :- .... ';+;l . .. . .""

."",; . .

.x

:x .. x x

x -

x

• xx

-

0.1 '---;1,!;5o..------;2do"'o-----;;;25~0;------;3~00

WAVENUMBER (cml)

5O.D

!z10.

i:! ~ 5.0 o u

is ~ 1.0

~ 0.5 CD «

ANGULAR FREQUENCY ud3rad /see) 2.0 2.5 3.0

NoCI 7°K

-0-

0.1=-...z...=--~12""0· -~14"'0---"16""0----:l

WAVENUMBER (eml)

Fig. 21.2. Band-mode absorption at 7 K due to natural isotopes in NaCl, 75.4% 35Cl and 24.6% 37CI. Dots, experiments (KLEIN and MACDONALD 1968); dashed curve and full curve, theory using different shell models, see

text. (MACDONALD et aI., 1969)

Fig. 21.1. Band-mode absorption at 7 K due to the lithium isotopes in LiF (solid circles, 50.8% LF; solid squares, 69.2% LF; solid triangles, 92.6% LF (natural abundance); +99.99% Lj?; cros

ses, 99.99% Li6). (KLEIN and MACDONALD, 1968)


constants nor the dispersion curves. The study of the isotope-induced spectrum, however, has given the information that certain critical points, as obtained from the shell model of Macdonald and Klein, have too Iowa frequency. (As a matter of fact, the breathing-shell model supplies frequencies much closer to the neutron spectroscopy data than either of the two shell models used by Macdonald and coworkers.)

The analysis of defect-induced absorption is, therefore, capable of giving valuable information about models for the host crystal. Among other matters,

CI)

.... i ::>

cD a: ~

>-.... (i; z '" .... z -0

'" a: '" .... I-<l <..> CI)

o a

KCI: TI+

SM

8SM

-if-

50 100

FREQUENCY

Eg THEORY

1m GEJw2 + il)

150

(em; )

200

Fig. 21.3a,b. Experimental and theoretical spectra of KCI:TI+. (a) Eg components; (b) TZg components. Spectra were taken at 15 K, the instrumental gain was the same for T2g and Eg components. The theoretical curves are for a 10% increase in the longitudinal force constants between the TI + and its nearest neighbours; the calculations were made using 0 K breathing-shell model (BSM) and 115 K shell model VI (SM) phonons. The theoretical spectra have been normalized to

reflect the experimental intensities. (HARLEY et aI., 1971) (Fig. 21.3b see page 292)


THEORY

SM

Vi ~

z ::>

ai 8SM It:

::!

)-

= V> Z ... .,1-

T2g EXPERIMENT

~

~

0 ... It: ... ~ ~ <l 0 V>

0 50 100 150 200

FREQUENCY (em-I)

Fig. 21.3b

the spectra depend very much on the transition matrix elements, that is, on the phonon eigenvectors.

On the other hand, the study of TI + -induced first-order Raman spectra by HARLEY et al. (1969, 1971; see also YACOBY and YUST, 1972; JUST and YACOBY, 1978) shows that the calculation of these spectra using the shell and breathing-shell models for KBr (cf. Fig. 25.3) shows barely any difference, while there seems to be practically no difference for KCI:TI+, as is apparent from Figs. 21.3 a and b where the appropriate r;.!(Eg) and r;~(T2g) Green functions from the shell (SM) and breathing-shell models (BSM) are compared with one another and with the Tl + -induced spectra. For the relative merits of the shell model versus the deformation-dipole model see KARO and HARDY (1975, 1976; HARDY and KARO, 1976) and PAGE (1976).

The systematic study of various defects in the same host by infrared absorption has been done on NaCI (WEBER and SIEBERT, 1968; MACDONALD and KLEIN, 1968; KLEIN and MACDONALD, 1968; MACDONALD et al., 1969), on KCI and KBr (TIMUSK and WARD, 1969; WARD and TIMuSK, 1970, 1972), on KI (SIEVERS, 1965a; WARD et al., 1975), and on CsBr and CsI (GENZEL et al., 1969; PRETTL and SIEP, 1971 a, b, c; BECKER, 1971). An analogous study

,..,

:... '-. iii c: CII

9 c:: o ~

Q:


of KCI using Raman scattering has been made by STEKHANOV and EUASHBERG (1963, 1964); STEKHANOV and MAKSIMOVA (1966); and of NaCI and KCI by MOLLER and KAISER (1969); see further references in Sects. 25b and c.

A general result seems to be that the frequencies of peaks in the absorption spectra caused by different defects (may be of the same valency) in the same crystal are essentially independent of the defect. An example is shown in Fig. 21.4. The suggestion is that these absorption peaks have their origins in density-of-states maxima or critical points. However, T.P. MARTIN (1971) in a theoretical study of the defect-induced absorption in Cs halides concludes that the absorption bands near 40 cm - 1 may be due to an incipient resonance (i,e. at a frequency where the determinant of 1 + go v, (23.15) below, is about to have a zero). Martin has used a breathing-shell model for the host lattice (MAHLER and ENGELHARDT, 1971) and has tried to fit the peak at 40 wavenumbers in CsI : K + (Fig. 21.5) by varying nearest-neighbor central and noncentral force constants at the defect, as well as the central force constants between the nearest neighbors of the defect. Best fits are obtained with only the central force constant between the defect and nearest neighbors unequal to zero. When this force constant is varied so as to fit the peak position, the latter is shifted into a

CsI:TI+

Csl:ln +

10 20 30 50 60 70 20 30 50 60

Frequency [cm -1 J Fig. 21.4. Defect-induced band-mode absorption at 6 K and phonon density of states in CsBr and CsI. Impurity concentrations are about 10- 2 mol %. The phonon densities are calculated from the

breathing-shell model with input prameters for 4.2 K. (PRETTL and SIEP, 1971 c)

2.5

294

'IE u

~ 0.3 QJ

'u :E ~ 0.2 u

c o E. 0.1 'o III .Q «

30


40 50 Frequency (cm')

Sect. 21

60

Fig. 21.5. Experimental (broken curves) and calculated (full curve) absoption coefficient at 4 K of 0.1 mol % in CsI for three different values of the nearest-neighbor central force constant change,

Lly= -3, -2, -1 x 103 dyncm-I. (T.P. MARTIN, 1971)

region of high (theoretical) density of states near the TA(X) critical point at 40 cm -1 which broadens the theoretical peak, as shown in Fig. 21.5. Martin concludes that the TA(X) critical point should be higher than calculated with the model, perhaps at 42 cm - 1.

WARD and TIMUSK (1970, 1972); TrMUSK and WARD (1969) found a concentration-dependent shift of critical points of KBr and KCl phonons, as observed in defect-induced spectra. Extrapolated to zero concentration, the positions of the observed critical points for various defects in KBr are listed and compared with theoretical data in Table 21.1. See also WOLL et al. (1968), WARD et al. (1975).

More complicated defects like molecules cause more complicated spectra. The interpretation of these spectra and the information so gained is more involved and will not be followed any further here. Some simple remarks can be found in Sect. 20g; the mathematical treatment is deferred to Sect. 26. As a

Table 21.1. Positions of the observed singularities in KBr crystals with different impurities at 11 K. The Na +, Sm + +, and Li + data have been extrapolated to zero-defect concentrations: the concentrations are otherwise as listed. The shell modell is that of WOODS et al. (1963). The van Hove singularities are assigned to a saddle point at <0.52, 0.52, 0), a maximum at <0.65, 0.35, 0.35), and two saddle points at <0.6, 0.6, 0) and <0.65, 0, 0) (in units of 2n/a), respectively. (WARD and TIMUSK

1972)

Mole % A B C D impurity (cm -I) (cm- I) (cm-I) (cm-I)

Li+ 0 74.71 75.1 85.67 89.56 Na+ 0 74.68 75.2 85.22 89.73 Tl+ <0.5 74.80 75.2 86.36 89.57 Sm++ 0 74.91 75.22 85.49 89.8 p- 0.001 74.84 75.53 85.67 90.17 OH- 0.013 74.60 75.5 85.6 90.15 0;- ~0.001 74.76 75.38 85.8 89.3 Mean 74.8±0.1 75.3±0.15 85.7±0.3 89.6±0.3 Shell model 70.7 71.3 83.0 88.0

Sect. 22 Lattice dynamics of impure lattices 295

matter of fact, the information contained in spectra induced by even the simplest defects can only be fully extracted after some mathematical investigations to which we will turn below.

Defects in alkali halides have been investigated very thoroughly; this seems to be because good models are available for the host crystals. The harmonic description of defects in alkali halides by a corresponding model has also reached a rather high degree of sophistication. In the semiconductors relatively little is known about harmonic perturbations, while rather interesting anharmonic effects occur. Among others, local-mode overtones are frequently observed; these are reviewed by NEWMAN (1969), see also Sect. 30j for further references.

22. Lattice dynamics of impure lattices a) Introduction: Molecular model - the nature of perturbations due to a

defect. In Sects. 20d and 21 we have given a very simple picture of the U center vibrating in its local mode. From the simple formula

w 2 =tP/M relating the local-mode frequency w to the force constant tP and mass M of the defect, we obtained the ratio wMw~ = 2 for the local-mode frequencies (isotope effect) in good agreement with experiment. However, if the host force constant is used for tP, one obtains values for the frequencies themselves which are too big by a factor of about 2 (JASWAL and MONTGOMERY, 1964; and others). Since the mass of the U center is well determined, one is led to the conclusion that in order to end up with the experimentally observed frequency, one has to assume a force constant cP which is about half as big as the corresponding one of the host lattice. This is indeed what has been found (see references in Sect. 20b) by using Green-function techniques which will be presented below. The success of this very simple model lies in the fact that the local-mode vibration is highly localized in the sense that the lattice, which cannot follow the rapid motion of the defect, essentially remains rigid in this vibration.

The defect can thus be regarded in first approximation as a simple point mass in a rigid box. This configuration is visualized in Fig. 22.1 a, where the outer ends of the springs are to be thought of as being fixed.

To give a more accurate account of the U-center local-mode dynamics, JASWAL (1965b) employed the "molecular model", an extension of the case so far considered in that the nearest neighbors are also allowed to vibrate along with the defect while the rest of the lattice remains at rest. This model, which is analogously shown in Fig. 22.1 b, seems to have been used for quantitative calculations first by LENGELER and LUDWIG (1963); DETTMANN and LUDWIG (1964); and LAND and GOODMAN (1965). There has been a recent revival of the molecular model (KRISHNAMURTHY, 1966; KRISHNAMURTHY and HARIDASAN, 1966,1969; HARIDASAN and KRISHNAMURTHY, 1969; STRIEFFLER and JASWAL, 1969a; SINGH and MITRA, 1970, 1972). The molecular model for the U center produces no dramatic changes in numbers or conclusions since the amplitudes of the neighbors' displacements are small compared to that of the defect, the ratio being typically a few percent (JASWAL, 1965.b). It does, though, result in a value for wMw~ which is less than 2 ..


a

z~ x

Fig. 22.1 a, b. Different models for a substitutional defect. In a molecular model the outer ends of the springs are thought of as being fixed. The vibrational displacements are thought of as being non-zero (a) only at the defect or; (b) at the defect and its nearest neighbors. Figure 22.1 b visualizes the mass and force-constant changes (LI m and Llf, respectively) as used in the connection with

the Lifshitz formalism

General arguments (KLEIN, 1968b, p.505ff.; PAGE, 1974; Sect. 30h) show that Einstein-oscillator behaviour (which gives w~w5=2) occurs only if the defect is vibrating in a strictly rigid lattice and that the deviation from Einsteinoscillator behaviour is due to the non-rigidity of the rest of the lattice. The fact that the U-center local mode demonstrates an approximate Einstein-oscillator isotope shift indicates that, disregarding anharmonic effects, the host-ion displacement amplitudes are indeed small compared with that of the U center. But, as the Einstein-oscillator behaviour is not perfect, the lattice does vibrate to a certain - even though small - extent.

The molecular model seems to permit rather exact estimates for vibrations which are well localized. With decreasing localization one would have to extend the size of the "molecule" to obtain reliable results. This has practical limitations, if the size of the molecule becomes too large. However, the U -center local mode is but one (three-fold degenerate) of the perturbed lattice vibrations. There are other vibrations which do extend far into the crystal and which are perturbed by the presence of the defect, even though the perturbations of force constants and masses due to the defect are limited to a small region around the defect. For these vibrations with frequencies close to or within the host-crystal phonon frequency bands the molecular model is bound to fail; different techniques for obtaining lattice dynamical properties have to be used.

The Lifshitz (Green function) formalism (LIFSHITZ, 1956; and references therein) has proven extremely successful. It allows the properties of the perturbed lattice to be calculated from those of the unperturbed lattice and from local changes. These changes which are brought about by the defect are twofold. First, the change in mass will alter the kinetic energy in an obvious way. Second, the interaction of the defect with the lattice will be different from that of the substituted ion or atom. Here one has to distinguish between short-range (e.g.


Born-Mayer) potentials and (Coulomb) potentials of long-range character due to altered charges at the defect. If the potential between the defect and any of its neighbors is changed - as it usually is - the neighbors will assume new equilibrium positions, i.e. the region surrounding the defect will distort. Even though there may not be a potential change between the defect and its neighbors other than the first, the potential expansion coefficients (force constants) will be changed since they are evaluated at different (namely distorted) equilibrium sites. These distortions will be the subject of the following section.

b) Lattice distortions - method of lattice statics. In order to place the rather general considerations of the previous section on somewhat more quantitative grounds a simple harmonic theory of lattice distortions around a defect shall be presented. The matter will be taken up again in Sect. 31 in a more general context including the effect of anharmonicities and defect concentration.

An impurity which is introduced into a crystal generally exerts some stress on its neighbors which results in a strain in its surroundings. The formulation of the general problem of calculating the strain which results from a force goes right back to early contributions by Lord KELVIN (1882); BOUSSINESQ (1885); and FREDHOLM (1900), who obtained the displacement due to a force applied at a point of an infinite isotropic solid (continuum) in terms of what is today called the static Green function. The continuum theory was further developed by ESHELBY (1956) and LIE and KOEHLER (1968).

In contrast to the continuum approach are methods which take into account the atomistic and thus discrete nature of a real crystal. For reasons of numerical effort, and because it was believed that the lattice far away from a defect would really behave like an elastic continuum, the crystal was treated as discrete in the defect region and as continuous far away from the defect. This approach was followed by MOTT and LITTLETON (1938); BRAUER (1952); and others. Here the problem of matching the displacements in the border regime arose. A discussion of the continuum and semi-continuum approaches was given by FUKAI (1963a) who also compared the theoretical predictions with experimental data (FUKAI, 1963b).

A discrete lattice method, the method of lattice statics due to MATSUBARA (1952), has been presented by KANZAKI (1957); HARDY (1960); FLOCKEN and HARDY (1968, 1970); and FLINN and MARADUDIN (1962). Although perfectly general in the beginning, this method has the drawback of generating a superlattice of defects as the consequence of a Fourier transformation at a later stage of the development. A computer simulation of the relaxation of a two-dimensional model lattice was made by CHUI (1975), to mention only one paper.

The ideas of the method of lattice statics are very simple and will be presented in the following. The Hamiltonian for the perturbed lattice, with the potential expanded around the unperturbed lattice sites, is

H=Ho+HA'

H 0 = t M Ii Ii + iP 1 U + t iP 2 U u,

1 4 HA =31 iP3 UUU + O(u ),


in obvious matrix notation; the last of these equations, for example, reads (L=I,K)

1 HA =-3' I <P3(L, 0:; I'., IX'; I'.', IX") u(L, IX) u(I'., IX') u(I'.', 0:")+O(u4 ) •

. LL'L" act' a"

As in Sect. 3, I and K are the cell and sublattice indices, respectively, which together describe the lattice site x(L) of an atom; IX is a cartesian index. Differently from Sect. 3, IX is written like an argument rather than a subscript in order to avoid confusion with other subscripts which appear below. We have left out the constant potential term. The linear term arises here from the fact that the defect will generally exert a force on the lattice resulting in a distortion such that the equilibrium positions of the atoms near the defect will not be those of the unperturbed lattice.

In a first approximation for this distortion we consider only the harmonic part, H 0, of the Hamiltonian. It is clear that the displacement u will consist of a static displacement (distortion) s of the equilibrium positions and of a vibrational displacement v around the new equilibrium positions,

u=s+v.

Substituting into Ha, one has (S=O)

Ha=tMvv+t~2vV

+~l V+~2SV

+~lS+t~2SS.

The last two terms give the first- and second-order changes of the constant part of the potential due to s, the first two terms have the usual form of a harmonicoscillator Hamiltonian. The remaining terms, which are proportional to the displacement v in first order, have to vanish since v is the displacement around the new equilibrium position which is shifted by s from the old one,

(~l +~2S)V=O.

Since this has to hold for any vibrational displacement, one finds 1

~1 +~2S=O,

s= _~"21 ~1' (22.1)

On the assumption that one knows the first and second derivative of the perturbed crystal potential at the unperturbed equilibrium sites one can calculate the distortion s of the crystal around the defect in the harmonic approximation. s is the response of the lattice to a given force field ~ 1; ~"2 1 therefore is a classical response function which will be seen in Sect. 22e to be the static, i.e.

1 Actually, that part of <1>2 which corresponds to translations (in terms of normal coordinates) has to be taken out of <1>2 before the inversion, see Sect. 31. If this problem is treated in the normalmode representation, as in Sect. 36a, the matrix <1>2 becomes diagonaL The zero eigenvalues of <1>2' which are due to the invariance of the potential under infinitesimal rigid translations, can then be exc1 uded,


zero-frequency limit of the phonon Green function. Hence (22.1) can be written as

s= _GST iP l ' (22.1')

The change of the constant potential term turns out to be

iP 1S+tiP2ss= -tiPl iPi l iP l = -tiPl GST iP l '

More generally one would have to expand the potential (or equivalently H) to all orders in s and v

s is now to be determined from the minimum of the free energy (FISCHER, 1959a, b). The details are given in Sects. 31 and 36 and are shortly summarized here. Neglecting the correlations between the displacements, which arise from thermal and spontaneous fluctuations, one obtains

00 1 I -, Sm<Pm+l =0 m~O m.

(22.2)

analogously to (22.1) which is recovered from the harmonic approximation of (22.2). (For an iterative procedure see, for example, NORGETT and FLETCHER, 1970; and references therein.) The effect of correlations is treated in Sects. 36a, where (36.25) and (36.26) are analogous to (22.2). Relating the static displacement s to the one-point function fl by (cf. (34.24))

s(L, oc) = I x(L, oc I A)(h/2w,,)1/2 fl (A)

" one can find corrections due to correlations from (36.28).

Example: Simple substitutional defects in alkali halides The method as summarized in (22.1') has been developed, further extended,

and applied to alkali halides by PAGE (1970), who followed a suggestion by ELLIOTT et al. (1968). Given a force field iP l (extending in Page's work only to the nearest neighbors of the defect), Page calculates the static displacement of any ion relative to one of the nearest neighbors of the defect by eliminating the force field. He also finds that the polarizability of the lattice ions leads to different displacements of ionic cores and shells (in the language of the shell model, see Sect. 27) (HARDY, 1962a; HARDY and LIDIARD, 1967), the difference being the larger the more polarizable the ions. Page's results for some ionic shells along a (100) direction are given in Fig. 22.2 for two alkali halides. Plotted is the displacement ss(nOO, x)/ss(lOO, x) of a shell at x(L)= [nOO] ro in the x direction relative to that of the nearest-neighbor shell. The model used by Page is the breathing-shell model. The lattice relaxation is seen to be very irregular in NaCI, which shows that the continuum theory cannot be applied close to the defect. The distortion does fall off as expected but does so less

300

1.0.

0..8

c 0 0.6 d x E ~

'" > :g 0..4 Qj 0::

0..2

0.

e


KI \ KI : -, \ ____ KI:. , \ , , , ,

\ \ , \ , ,

\

~ \ \ \ \

\. .. , Neighbor

e

\ \ , , \ , , \

NaCI NaCI: -

---- NaCI:.

, , \ , , , , ,

\ , \ \ \

~ \ \ \ \ \ \ \ \

e~ \e\ 1:>.,

'~

Relative shell relaxations along <10.0.) directions

Sect. 22

Fig. 22.2. Relative static relaxation of the electronic shells (see text) of atoms in a < 100.> direction as a function of the distance from a substitutional impurity. Full curves, substitution of a halide

ion; dashed curves, substitution of an alkali ion. (PAGE, 1970. and private communication)

slowly than generally assumed. On the assumption of short-range and weakly anharmonic potentials (and/or small relative static displacements) distortioninduced relative force constant changes

Ll cPss((n + 1)00, x; nOO, x)/ Ll cPss(200, x; 100, x)

can be calculated from the relative displacements between neighboring atoms. The results of such a calculation are given in Table 22.1. The quantity LlcPss(OOO, x; 100, x) is influenced by the additional perturbative potential due to the defect. This is an independent quantity. Note that the force field cP 1 has been eliminated and that the results of Fig. 22.2 and Table 22.1 are thus independent of the perturbation cP l' The defect-(not distortion-)induced perturbations in cP 2

have been neglected in these calculations as far as they extend beyond the nearest neighbors. There is a relation like (22.20') below with s substituted for u. The implication of the results of Table 22.1 on the dynamics of defective lattice will be discussed in Sect. 22e.

c) Equation of motion of the perturbed lattice. The following sections will be concerned with harmonic perturbations and their influence on lattice properties.


Table 22.1. Relative force constant changes

Aqj([n, 0, 0], x; [n-1, 0, 0], x)jAqj([2, 0, 0], x; [1, 0, 0], x)

between shells due to relaxation around a defect which is substituted at a positive ( +) or negative (-) ion lattice site (x([n, 0, 0])=0) in two alkali halides. See text also. (PAGE, 1970 and private

communication)

Host Defect n=2 n=3 n=4

NaCI + 1.0 0.48 0.08

1.0 0.10 0.18

KI + 1.0 0.60 0.26

1.0 0.33 0.27

A knowledge of the phonon propagator in the defective lattice is essential for the calculation of effects caused by phonons. In the previous chapters the propagator was formulated in reciprocal (q) space, but here a real-space representation shall be employed since the periodicity of the crystal is destroyed by the presence of impurities.

To set down the notation for the following sections, one starts out from the equations of motion for the unperturbed lattice,

(22.3)

this is a matrix notation for

I [<1>o(L, a; E, a')-w2 Mo(L, a) bLL, baa] u(E, a')=O. L'a'

The quantities have the same meaning as in Sect. 3, the index zero refers to the ideal-lattice quantities. The mass matrix is diagonal and does not depend on the cell index nor on the cartesian coordinate a. However, the indices have been kept for the convencience of matrix notation and because the mass matrix in a perturbed lattice will depend on the cell index. The notation is otherwise as in Sect. 22b. Equation (22.3) has solutions if

det(<<Po - w2 M 0) = 0

holds. The solutions are then given by, cf. (10.1),

U=loQ,

which is the matrix representation of

u(L, a)= I Xo(L, alA) Q(A) with ..

(22.4)

The symmetry properties of lo are considered in Sect. 22d. The matrix lo contains the eigenvectors solving (22.3). The eigenvalues wi = w2 (q, j) correspondingly are elements of a diagonal matrix WOo The solved eigenvalue equation (22.3) then reads

(22.5)

302 Lattices with point defects

The eigenvectors are complete and orthogonal 2 (not unitary),

M6/ 2 10 16 M6/2 = 1,

16 M olo=l, or, in components,

M6/ 2 (L, 0:) L: Xo(L, o:IJe) X~(l'; 0:' I Je) M6/ 2 (L; o:')=bLL,b",,,,,, A

L: x~(L, o:IJe) Mo(L, 0:) Xo(L, 0:1A')= bu '. L",

Sect. 22

(22.6a)

(22.6b)

Similarly, the equation of motion for the lattice with defects is written as

(IP-w2 M) u =(IPo _w2 Mo +LlIP-w2 LIM) u

=(IPo _w2 Mo + V) u =0, (22.7)

where the changes in the force-constant matrix IP and mass matrix M with respect to the ideal host lattice matrices are denoted by LlIP and LIM, respectively.

(22.8)

is the perturbation matrix. The solutions of (22.7) are formally given in terms of perturbed eigenvectors

X(L, o:IJe) which are normalized according to

Ml/211t Ml/2 = 1,

ItMI=I.

(22.6c)

(22.6d)

In practice, the huge matrix IP - w 2 M as it stands cannot be diagonalized. However, the Lifshitz formalism and symmetry considerations reduce (22.7) to a tractable problem.

Example: A simple defect in a lattice with the rocksalt structure

A substitutional defect in a cubic crystal with inversion symmetry may serve as an illustration. It shall be assumed that only the longitudinal force constants between the defect and its nearest neighbors are changed; these changes are depicted in Fig. 22.1 b. For the host crystal we assume a model that couples ions to one another by harmonic force constants.

The defect site shall be taken to be at the origin, x(L) = [000], simply denoted by 0. If LI m and LlJ denote the change in the mass of the defect and the change in the longitudinal force constant between the defect and its nearest neighbors, then LIM and LlIP have the following non-zero elements,

LlM(O,o:)=Llm,

LI cJ>(0, 0:; 0, 0:) = 2L1f,

LlcJ>(O, x; 100, x)=LlcJ>(O, x; 100, x) = LlcJ>(O, y; 010, y)= ... = -Llf,

LlcJ>(100, x; 100; x) = LlcJ>(100, x; 100, x) = LlcJ>(010, y; 010, y)= ... =Llf (22.9)

2 The complex conjugate, the transpose and the adjoint (hermitian conjugate) of a quantity shall be denoted by a star (*), a T, and a dagger (t), respectively.


As cP and cPo have infinitesimal translational symmetry, so must ilcP=cP-cPo, which is easily verified. If we arrange the rows and columns of the defect matrix in the following order of the element indices (with notational convenience for the following)

e 100 100 0 010 010 0 001 001 ... ),

x x y y y z z z

then A 0 0

0 A 0 0 V=

0 0 A (22.10)

0 0 with

CAl-CO' Am -ilf -AI) A = -ilf ilf o .

-ilf 0 ilf

d) Symmetry considerations. In general, there are some normal vibrations in the lattice which are changed by the presence of the defect and others which are not. This depends on the structure of the defect and the host lattice. Symmetry arguments show which of these vibrations are perturbed (BIRMAN, 1974b). If one considers the simple example of a single isotopic defect (il cP = 0, change in kinetic energy only), then the only vibrations which are changed are those which involve motion of the defect. In cubic crystals in which the defect is at a center of inversion (as in the example), these are the vibrations of 1;.5 (Tl u) symmetry. In these crystals the vibrations fall into two categories, even and odd parity. These are defined by the properties of the displacements under inversion,

X(L, oc I Je) = X(x(L), ct I Je) = ± X( - x (L), ct I Je) for {Odd} symmetry. even

Equation (22.7) describes changed and unchanged vibrations as well. Application of group theory separates this equation into at least those two uncoupled sets which describe perturbed and unperturbed vibrations. As a matter of fact, decomposition of matrix equations into different irreducible representations of the point group, even though not essential for developing the general theory, is of great practical importance, because the reduced size of the matrices facilitates numerical computations.

Formally, the decomposition of the equation of motion is accomplished by a unitary transformation, F,

F(cPO-W2MO+ V)Ft Fu=O.

F has elements

F(r, r, slL, ct);

here r denotes the representation, r is the row of the representation, and s the "multiplicity" or "realization" index. We will see below that F is undetermined


with respect to the multiplicity index s. The ionic displacements according to a given representation r are then given by

u(r, r, s) = L F(r, r, s I L, oc) u(L, oc). (22.11) La

In order to determine the possible symmetries of the perturbed vibrations, one looks for those representations for which the corresponding perturbation matrix FVFt does not vanish. V later on will be assumed to have non-vanishing elements only in a limited subspace, in the so-called "impurity" space. So we need to perform the unitary transformation within this impurity space only.

In general, the transformation matrix F need not be unitary. However, unitarity assures normalization of the symmetrized displacements if the real displacements are normalized.

The matrix F consists of a sum of operations R of the point group. Acting on a displacement of an atom at a given lattice point R rotates the displacement vector and the corresponding lattice site. This can also be looked at as a rotation of the coordinate system. Since the wavevector space is orthogonal on the real space, a rotation of the real-space coordinate system is the same as the inverse rotation of the wavevector-coordinate system. If the point-group operation R is applied to the eigenvector 10 we have with x=x(L) and RT =R- 1 and up to a phase factor

lo(xlq,j)=R- 11o(Rx I Rq,j) =(MO,,,N)-l R- 1 e(KIRq,j) ei(qTR-I)(Rxl,

using the representation (22.4), and thus

R- 1 e(KIRq,j) =e(KI q,j).

If, in particular, the inversion is an operation of the point group one has

e(Klq,j)= -e(KI-q,j) and

lo(xlq,j)= -l~(xl-q,j).

In comparison with (34.3) this suggests a choice of the phase factor (J = -1. The notation mainly used in this article is that of LOUDON (1964a). A

number of different symbols are used in the literature; the ones most commonly used have been compiled by SLATER (1963). Slater's collection, slightly augmented, is given in our Table 40.2. Character tables are given in nearly every treatment of group-theoretical aspects of crystal physics. The irreducible representation of photons is 1;:5"(115) for the group 0h(Td). Phonons of 1;:5"(1;:5) symmetry are, therefore, infrared-active. Space-group reduction of

I;:5XI;:5"=I;:++I;:!+I;:!+~! ~J

I;: 5 X I;: 5 = I;: +I;:2+I;:5+~5 (Td)

gives the sum of the irreducible representations for which phonons are Ramanactive.

Example: The simple defect in a cubic lattice


Taking up the example of Sect. 22c, the defect with changes in mass and force constants, the lattice equation of motion (22.7) will be decomposed into uncoupled equations according to the irreducible representations of the group 0h = Td X I of the crystal in the example. The matrix elements of the transformation matrix F for the group Td are given, e.g., by SLATER (1963, p.358). V, from (22.10), has non-vanishing elements only in the 9-dimensional subspace of the impurity and its 6 nearest neighbors. So we need to perform the transformation within this so-called "impurity space" only. The required elements of F are listed in Table 22.2. For the sake of completeness the elements of F are given for the total 21-dimensional subspace of the impurity and its 6 nearest neighbors. (The difference between the dimensionalities is due to the fact that in the example only 6 of the 18 degrees of freedom of the neighbors involve the perturbation). The corresponding displacements u(r) appropriate to different irreducible representations are pictorially shown in Fig. 22.3. An analogous list of elements of F for the group Td is given in Table 22.3 with the corresponding displacements shown in Fig. 22.4. Similar, but more extended tables and those

Table 22.2. Elements n . F(r, r, sll, K, IX) of the transformation matrix F for a substitutional defect and its nearest neighbours at site x (I, K) in a lattice with the NaCl structure (group O~). r is the representation, r the row (partner), s the multiplicity, and n a normalization factor. (For third-

nearest neighbours see Table 22.3). The corresponding displacements are shown in Fig. 22.3.

x(/, K) [000] [100] [100] [010] [010] [001] [001]

IX x y z x y z x y z x y z x y z x y z x y z r r s n

r.+ 1 V6 1 -1 1 -1 1 -1

r;~ 1 VU 2 -2 -1 1 -1 1 2 2 -1 1 1 -1

r;~ 1 2 1 -1 -1 1 2 2 -1 1 1 -1 3 2 1 -1 -1 1

1;~ 1 2 1 -1 1 -1 2 2 1 -1 1 -1 3 2 1 -1 1 -1

r;-" 1 1 1 1 2 V2 1 1

3 2 1 1 1 1

1 1 1 2 2 V2 1 1

3 2 1 1 1 1

1 1 1 3 2 V2 1 1

3 2 1 1 1 1

1;-" 1 2 1 1 -1 -1 2 2 -1 -1 1 1 3 2 1 1 -1 -1


r* ~ ~:+ r,;t7¥~ r"Lf, r,~.f-/

r,;:~t r,'~f r,~

r,;+ r,;i_f _ r,,; • .{:

5 = 1 5=2 5=3

r,;+ I;~~r~ I;;~+

5 =1 I 5= 2 5=3

I,;" t ~ --- I;;~: +. I;;~

5 = 1 5 = 2 5=3 t-~

r,;;~ t:: r,f r,;~ a

r,;~f- ~f- -{-'r 5= l' 5= 2' b

Fig. 22.3a, b. Displacements of a substitutional impurity and its six nearest neighbours in a crystal with Dh symmetry according to the irreducible representations of the group Dh. The realizations of the irreducible representations are not unique: (a) shows the realizations according to the matrix elements given in Table 22.2; (b) shows another possible realization for the first row of the repre-


Table 22.3. Same as Table 22.2 but for a lattice with the ZnS (or diamond) structure (group Td). The corresponding displacements are shown in Fig. 22.4. The elements of F for a defect and its nearest neighbours in the CsCI structure (group O~) and for a defect and its third-nearest neighbours in the NaCI structure (group O~) can be obtained by adding to the table the elements F(r, r,

sl-(/, K), IX) = ±F(r, r, sll, K, IX) for the odd (+) or even (-) parity vibrations.

x(l, K) [000] [111] [11I] [111] [II1]

IX x y z x y z x y z x y z x y z

r r s n

I;. 02 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1

I;. 2 1 v14 2 -1 -1 2 1 1 -2 -1 1 -2 1 -1 2 i!8 -1 1 1 -1 -1 -1 1 1

I;.s 1 1 1 1 2 2 1 1 1 1

3 vB 1 1 -1 -1 -1 1 1 -1

1 1 1 2 2 2 1 1 1 1

3 vB 1 1 -1 1 -1 -1 1 -1

1 1 1 3 2 2 1 1 1 1

3 vB 1 1 -1 1 1 -1 -1 -1

12s 1 Vi -1 1 1 -1 1 1 -1 -1 2 iii 1 -1 -1 -1 -1 1 1 1 3 i!8 -1 1 1 1 -1 -1 1 -1

for further site symmetries are given by LUDWIG (1964); DETTMANN and LUDWIG (1964).

As stated above, the decomposition of the displacements (and matrices 4>0' V, etc.) into realizations of the irreducible representations is not unique. For example, instead of the set

u(Frs, 1, 1)=u(0, x),

1 -u(Frs, 1,2)= v'2 [u(100, x)+u(lOO, x)]

(22.12)

as it appears in the example, one could just as well have taken

1 -u(Frs, 1, 1') = v'3 [u(O, x) + u(100, x)+u(100, x)],

1 -u(Frs,1,2')= v'3 [u(O, x)-u(100,x)-u(100, x)],

pictorially shown in the bottom of Fig. 22.2, or any other combination of u(Frs, 1, 1) and u(Frs, 1,2).

308

5 = 1

I25,1


I15,l

I e I --/

1£.-/---'" 5 = 2

/

1L/---~5=2

ie_ 5 = 2

I15,1

/ /

/ /

I e

I

I e . I

/ ~-

I e I

/"'-:- -

I I e I

~:--/'

Sect. 22

5=3

Fig. 22.4. Displacements of a substitutional impurity and its four nearest neighbors in a crystal with Td symmetry according to the irreducible representations of the group Td

Using Table 22.2, one finds for the defect matrix (22.10)

V(l~+)

V(I;:!, 1) 0 V(I;:!, 2)

FVFt= V (I;: 5, 1) 0

0 V(I;:5,2)

V(I;:S,3)

0 0


with

V(I~+)= (L1t I ~), (22.13 a)

V(~!, 1)= V(~!'2)= (6' I ~), (22.13 b)

(2L1f-W2 L1m -V2L1f

V(~5,1)=V(rl5,2)=V(r15,3)= -V2N N

o (22.13 c)

Of the ten possible representations of 0h' only three result in non-zero perturbation matrices for the particular choice of the defect and the perturbations induced by it This means that phonons of all symmetries except ~+, ~i, and ~5 are unperturbed, For an isotopic defect (L1f =0), only vibrations of ~5 symmetry are affected as mentioned above,

e) Lifshitz method for the solution of the equation of motion - localization of perturbations, To solve the complicated and large set of equations (22,7) one takes advantage of the fact that one knows the eigenvibrations of the unperturbed crystaL One factorizes (22,7)

(<Po-w2Mo)(1+GoV)u=0 (22.14) where

(22.15)

is the classical (Lifshitz) Green function of the ideal lattice. Some of the literature uses a slightly different notation. In this notation (22.5) is written as a standard eigenvalue equation,

(Mo 1/2 <Po Mo 1/2) (MlP Xo) - (M6/2 Xo) w~ = 0

== cPoXo - XoW5 =0. Correspondingly, (22.7) reads

(cPo -w2 1 +L1cP-w2 L1M) u=O with

Finally, one has

Go = (cPo - w2 1)-1 =M6/2 (<Po _w2 M O)-l M6 /2 = M6 /2 GoM 6/2

and V=MOl/2 VMOl/2.

Occasionally - V is used instead of V and - Go or - Go instead of Go or Go, respectively.

The Lifshitz Green function of the ideal as well as of the perturbed lattice (to be defined in Sect 23) plays a central role in the representation of the properties of pure and perturbed crystals. Equation (22.15) seems to have been established first by Lifshitz, but it remained unrecognized outside Russia for about ten years


(see LIFSHITZ' review article in 1956 with references to his early work). In the meantime LAX (1954); KOSTER and SLATER (1954a, b); and KOSTER (1954) derived the same expression for the electronic defect problem. Early work with Green functions on the lattice dynamics of perturbed lattices was done also by MONT ROLL and POTTS (1955). It was not until the work done by BRODT and VISSCHER (1962) and KAGAN and IOSILEVSKII (1962) that quantitative calculations of defect properties in crystals (DAWBER and ELLIOTT, 1963a, b) were stimulated.

The condition for solubility of (22.14) is the vanishing of the determinant

0= det [(4)0 - w2 Mo) (1 + Go V)] = det(4)o _w2 Mo) det(1 + Go V)

which is valid if either the first or the second determinant on the right-hand side is zero. The first determinant being zero is the condition for obtaining vibrations unaffected by the defect (if there are any), while the second will give the perturbed frequencies (since Go and V depend on frequency).

One could have gone a slightly different way from (22.14) on. Since one is interested in perturbed vibrations, one could have divided (22.14) by 4>0-w2 Mo since this term has zeros at unperturbed frequencies only. Then one would have been left with the equation of motion

(22.16) and with

det(I+Go V)=O (22.17)

as the solubility condition for the perturbed phonon eigenvalue-type problem in both cases. Equation (22.17) still seems to require the evaluation of the determinant of a very large matrix, even if reduced in size by symmetry. However, one will see in a moment that things will become much simpler. Equations (22.16) and 22.17) are extremely easy to handle in the case in which the perturbation V is limited (as in the example) to a small region of the crystal, to the so-called "impurity space". First the validity of the assumption that there is indeed a small impurity space will be investigated, and then the consequences will be discussed.

For the defect space to be small, one first has to restrict oneself to very few defects in the crystal, preferably to a single one, or one defect pair, etc. Section 28 explains how the results can be generalized to several defects. Then the perturbation matrix V = Ll4> -w2 LIM of that single defect (or defect pair) may have only a few non-zero elements: Clearly, LIM is localized. For Ll4> to be localized, one first has to assume that the change in potential is localized. For a shortrange potential, e.g. of Born-Mayer type, this is clearly the case. One also has to assume that the defect has the same charge as the original host atom. Otherwise there will be the long-range Coulomb potential change. See also Sect. 27, where this restriction will be slightly relaxed.

Even if there is no Coulomb potential change, there are in general longrange perturbations induced by the lattice distortion around the defect. Equation (22.1'),


shows that even though the first-order potential change may be localized, the resulting distortion will not be, since GST has long-range character. In fact, inspection of Table 22.1 shows that the change in force constant between the third- and fourth-nearest neighbors (along the cubic axes) of a negative defect in NaCI is even larger than the change between the third- and second-nearest. This is due to the fact that the distance between the third and the fourth atoms has changed more than between the third and second atoms, cf. Fig. 22.2. The reduction of the impurity space to just the defect and its nearest neighbors is, in principle, very doubtful. Only in the case where the inhomogeneous strain does not extend into the crystal to any appreciable extent, can one hope that the force constants are essentially unchanged in the crystal except for a small region around the defect.

In principle, the polarizability of the lattice ions or atoms leads to effective long-range forces (Sect. 3); defects therefore induce long-range changes even though, in terms of the shell model, the change of force constants between shells may be localized. We will show in Sect. 27 how this problem can be overcome.

One now divides the crystal space into the small "impurity space" (index I) and the "remaining region" (index R). Lower-case letters will be used for the impurity subspace matrices; V= l'iI and go=GoIl then denote the defect and Green-function submatrix, respectively, in the impurity space. Equation (22.16) is then written as

[ (~ 0) + (GOIl 1 GOR1

G01R ) (V GORR 0 ~) ] (::) =0 (22.18)

or (l+go v)uI =O, (22.19)

UR = -GOR1 vU1 (22.20)

=GOR1(gO)-1 u1· (22.20')

In expression (22.19) one has arrived at an equation which is to be solved in the (very small) impurity space only. This is the great advantage of the Lifshitz formalism. Once one has found the eigenvectors in the impurity space, those in the remaining space can be calculated by simple matrix algebra according to (22.20).

The determinant of 1 + Go V, (22.17), written in the form of (22.18), can be expanded to give

(22.21)

in accordance with (22.19). Equations (22.12) and (22.13) have demonstrated that the application of

group theory helps to reduce even further the size of the matrices go and v, thus making numerical computations more tractable.

OITMAA and MARADUDIN (1969); OITMAA (1970, 1971 a, b); LITZMAN et al. (1972); and LITZMAN and R6sZA (1965) present different methods for an evaluation of the secular equation (22.21) for the case of far-extended defects. LITZMAN and his coworkers (LITZMAN and BARTUSEK, 1971; LITZMAN and


LACINA, 1972) also point out that not the order of the defect matrix but its rank determines the minimum size of the matrices entering into the secular equation.

For localized modes of sufficiently high frequency one may expand the denominator of the Green function (GUNTHER, 1965a; MATTHEW, 1965; see also the moment expansion by LANNOO and DOBRZYNSKI, 1972)

go=lo(W~-wU)-llb

For more rapid convergence it may be useful to expand not around wi/wl =0 but to write

with w chosen to be some kind of average lattice frequency. The zero-order expansion term in both expansions is diagonal and equal to Mol, see (22.6a). In the linear case (Einstein oscillator limit in which only the defect vibrates whence the matrix relations reduce to those between scalars) one then finds

with

F or the Einstein oscillator extreme (w~ ~ < wn») one finds

2 2 -2 1 v=L1cp-wL Llm=(wL -w )mo~--go

or

wl(mo + L1m)=.1 cp+w2 mo.

Here .1 cp is the defect-site force-constant change; w2 mo then has to be equal to the defect-site force constant (cf. also KLEIN, 1968a). A choice of w between the transverse optical and longitudinal optical host-lattice frequency seems to be rather physical, and a choice of w2 = < ( 2 ) makes the first-order expansion term of the Green function (in the linear case) vanish.

Example: An isotopic defect in an N aI lattice The solutions of the secular equation (22.21) will be discussed for an even

simpler defect than before, namely an isotopic defect (.14> = 0) at site x(L) = O. In


this case (22.21) becomes linear,

L Xo(O, a I .ic)(wi - ( 2)-1 X~(O, a l.ic) ;.

=go(O, a; 0, a)= _v- 1 =(w2 Am)-l =(w2 8mo)-1 (22.22)

8 = A mlmo being the fractional mass change. The Green function, go, has poles at the frequencies of the unperturbed

frequencies. For a real crystal these frequencies are very densely spaced. To visualize the solutions of (22.22), all poles of go except at a few frequencies are omitted. In fact, the coarsest possible non-trivial mesh of vectors in the Brillouin zone, equivalent to a periodicity volume of 8 elementary cells is taken. The eigenvectors Xo(.ic) and eigenfrequencies w;. are calculated from a model (breathing-shell model, Sect. 3) of NaI with the defect at a negative-ion site. The eigenfrequencies of this lattice are enumerated and marked at the top of Fig. 22.5. In the lower part of Fig. 22.5 go(O, a; 0, a) and -llv for some arbitrary

~Acoustical band Gap wtj

Optical band 2 3 56 7 89

, , : : WG ' , , WL , , , , ,

I , I

1

I ! \ !

Or I ! I , Nal: -\ , , I

\ , , \ : , I : I \ \ : I \ I I

I i I Q 51- I I I

\ I I

\ ' £=13.5 \ i ,: I

'" I I 3.5 .1013 " 3.0 ' .. ! I -- - ..J. -_.j.. .L I I I a -.

I 0.5 I 1 I 1.5 2.5 :

Ii I

11 r--, , -----_. I I _----- ------

! , -- ---------, --, ..-- -, .",.- --' .... '£=-0.75 ..----C " --5- " ......... (.-0.3 -0.

-1.0

" , 1/

/ /' I , /

I , I ,

I , r I I

I I I

I I

I I I

Fig. 22.5. Graphical solutions of the secondary equation (22.22) from the equations of motion in a perturbed lattice (schematical). Thick full curves, the unperturbed Green function, go, versus fre· quency, OJ. The thin full lines mark the position of nine representative unperturbed eigenfrequen· cies. The dashed curves are -l/v versus OJ for three values of the mass changes (6 = A m/m). The perturbed eigenvibrations are given by the positions of intersections of go and -l/v. These positions are marked by the dotted lines. For heavy defects (6)0) the perturbed eigenfrequencies are shifted to lower frequencies, as indicated on the top of the figure, and inversely for light defects.

The frequencies OJG , OJH , and OJL are explained in the text


values of LIm are plotted as a function of frequency. The intersections of go with -l/v mark the solutions of (22.22) or (22.21) and thus give the frequencies of the perturbed vibrations. The positions of the perturbed frequencies are also marked in the upper part of Fig. 22.5. All frequencies are slightly shifted, the shift being positive (negative) for light (heavy) defects, as is intuitively clear. For decreasing perturbation the dashed lines ( -l/v) move out of the frame of Fig. 22.5, and the perturbed vibrations merge into the unperturbed ones. Since in real crystals the poles of go are -very close together, the shifts of the perturbed frequencies are actually extremely small. In a real crystal the vibrations with frequencies denoted by W G , W H, and WL appear in regions with large energy separations and thus usually show sizable shifts. For light defects (LI m < 0) WG and W L emerge out of the acoustical and optical band, respectively; for a heavy defect the frequency WH is split out of the optical band. These frequencies are well localized in the frequency region outside the bands. As mentioned in Sect. 20d, the corresponding vibrations are called localized modes; the vibrations with frequencies wG and WH

which fall in the gap between the bands are often called gap modes. A more quantitative discussion of the dependence of the frequencies Wu WH, and wG will be given in Sect. 23 d.

Figure 22.5 shows that there is a gap mode for any mass change (or forceconstant change). In actual calculations the range of values which LIm (or Llf) may assume to produce a gap mode is rather restricted and may even have zero extent, for reasons to be discussed below (Sect. 23 d) in connection with complex Green functions.

Taking up the slightly less simple example from Sect. 22c and d, one sees from (22.l3c) that the calculations of phonon frequencies of I;.5 symmetry requires the evaluation of a 2 x 2 rather than a 1 x 1 matrix. However, this changes the qualitative conclusions only in so far as the poles of the dashed lines in Fig. 22.5 move about the frequency axis, thus leading to partially positive and partially negative frequency shifts. In this case one has for r= 1, cf. (22.12),

with

and

gij= L: F(r15 , 1, ilL, oc) go(L, oc; E, cl) F(I;.s, 1,j I E, oc') LL'M:

gll = go (000, x; 000, x)

g12=g21 =y2go(OOO,x; 100,x)

g22 = go(100, x; 100, x) + go(lOO, x; 100, x)

V= (2L1f _w2 LIm -y2L1f

-y2 L1f) Llf '

cf. (22.13c); then (22.21) becomes

(1-w2 LIm) gll +Llf(g22 -2V2g12 +2g11)

- w2 LIm Llf(gll g22 - gi2) =0. (22.23)

Sect. 23 The Green function of the harmonic perturbed lattice 315

The defect matrix for phonons of ~+ and ~i symmetry (in fact, of all symmetries which do not involve motion of the defect) is independent of frequency. The secular determinant indeed is linear, and the solutions would be given by intersections of curves similar to go in Fig. 22.5 with the straight lines -l/v = - 1/ L1f A weakening in force constants (,1 f < 0) would result in shifts to lower frequencies and vice versa. A related example will be found in Sect. 23 where the present example is also taken up again.

LEDERMANN (1944) has shown that an eigenfrequency of a perturbed lattice cannot shift from the corresponding one of the unperturbed lattice by more than its distance to the neighboring unperturbed frequency. This is nicely demonstrated in Fig. 22.5. However, Ledermann's arguments seem to hold for the poles of the Green function only for a given representation. By suitably choosing the defect properties, one can see that the shift of a perturbed phonon (here the local mode) is, in fact, larger than the spacing to the next unperturbed frequency (labelled No.9 in Fig. 22.5).

23. The Green function of the harmonic perturbed lattice. The interpretation of phonon spectra in the foregoing Chaps. C and D has been in terms of the Green function. Since the multi-phonon Green function can in principle be obtained from the one-phonon function, the latter is the key quantity. As with the pure-crystal spectra, one can interpret the perturbed-crystal spectra in terms of the Green function of the perturbed crystal. The knowledge of the Green function of the perturbed lattice is, therefore, of fundamental importance for the understanding of the defect-induced spectra. To begin with, the influence of the harmonic properties on the lattice Green function will be discussed, anharmonic effects will be deferred to Sect. 30.

The calculation of the Green function is straightforward but nevertheless lengthy if carried out in some detail. In order to demonstrate the methods, a perturbation matrix of dimension one is often used (as in the example of Sect. 22e), in which case the involved matrix inversions become trivial. We refer to this as to. the linear case.

The Green function of the perturbed lattice is introduced in Sect. 23 a. The interpretation of that function as the propagator of a lattice phonon which is successively scattered at a defect leads to a scattering (T) matrix which is also introduced in Sect. 23 a. After a comment on the analytical continuation of the Green function into the complex frequency plane in Sect. 23 b various types of resonances are discussed in Sect. 23 c. Before entering the discussion of the properties of resonances in Sect. 24, we present in Sect. 23d an eigenvalue treatment of the pseudo-eigenvalue problem as formulated in (22.16) or (22.19).

a) Real Green function and T matrix. Analogously to (22.15), the Green function of the perturbed lattice is defined as

G = ((j) - w 2 M) - 1 = ((j) 0 - w 2 M 0 + ,1 (j) - w 2 ,1 M) - 1

=(G0 1 + V)-l

=(l+Go V)-l Go=Go(l+ VGO)-l

=Go-GoTGo (23.1 )


with the T matrix defined as

T= V(I+Go V)-l =(1 + VG O)-l V= V- VGv,

T and G are very similar in structure; one has

TGo=VG.

Note that G shows resonance behaviour whenever T does, and vice versa.

Sect. 23

(23.2)

(23.3)

The T matrix defined by (23.1) is the lattice-dynamical analog of the quantum-mechanical scattering matrix (MESSIAH, 1962; for example) defined by (cf. (22.16))

(4)0-w2 Mo+ V) u=O,

(4)0 _w2 Mo) u= - Vu

= -Tuo,

where uo is the solution of the homogeneous equation. The formal solution of

u=uo-(4)0-w2 M O)-l Vu=uo-Go Vu

for Vu in terms of uo is equivalent to (23.1). Go has poles at the unperturbed phonon frequencies with the eigenvectors

forming the corresponding residua. Similarly, the poles of G mark the perturbed phonon frequencies. Because of the matrix (1 + Go V)-1 in (23.1) the poles of G are at the zeros of the determinant det (1 + Go V). This was exactly the condition of solubility, (22.17).

From (23.1) one finds a Dyson equation for G which can be solved by iteration,

(l+Go V)G=Go G=Go-Go VG

=Go-Go VGo+Go VGo VGo- + .... (23.4)

A way of interpretation of (23.4) is that the phonon propagates through a defectfree region of the crystal (Go, first term) until it is scattered by the perturbation, V, then continues in the perturbationless region (second term), possibly is scattered again (third term), and so on. The T matrix contains all successive scattering processes.

The Green function of the perturbed crystal, G, can be expressed in terms of the one of the unperturbed crystal, Go, and the changes, V, introduced by the defect. Since V has non-zero elements in the impurity space only, the same is true for the matrix T. From (23.1) and Sect. 22e one obtains

with

T= (~: ~:) = (~ ~) [(! ~) + (~::I Z:::) (~ ~) r 1 = (~ ~) (23.5)

(23.6)

Sect. 23 The Green function of the harmonic perturbed lattice

and similarly

(and analogously for Go) with

Gn =g=(I+ go V)-l go=go- go tgo,

GIR =(1 + go V)-l GOIR =(1- got) GOIR '

GRR =GORR -GORI tGOIR •

From here one also finds

t=v-vgv

in analogy to (23.2).

317

(23.7)

(23.8 a)

(23.8b)

(23.8c)

(23.6')

b) The complex Green function. The pure-crystal Green function, (22.15),

Go=Xo((J)~-(J)21)-1 X6,

exhibits rapid variations as a function of frequency with poles along the real frequency axis at the eigenfrequencies of the unperturbed crystal. Introduction of a defect shifts these poles by small amounts as discussed in Sect. 22e. Because of damping in a real system the poles of the retarded Green function are removed from the real frequency axis into the lower half of the complex frequency plane. To avoid divergences along the real frequency axis for harmonic (undamped) systems a small artificial damping is introduced. Another way of expressing the same fact is to say that complex frequencies (w + il::) slightly above the real frequency axis are considered (adibatic switching-on). In any case, the effect is that as long as I:: is small but sufficiently large compared to the distance between two adjacent poles (a condition that can be fulfilled by sufficiently increasing the size of the crystal thus decreasing the distance between poles) the Green function becomes a complex quantity,

G= [<I>-(w+il::)2 M]-l

(23.9)

and the variations in G are smoothed out if w in this expression is taken real. Assuming for somewhat more generality a damping 'Y;. for a mode with frequency W;. G has the form

G= I x(A)[w;-(w+i'Y;.)2]-1 Xt(A) (23.10) ;.

with poles at w= ±w;.-i'Y;..

Rigorous results are obtained in chapter G. The susceptibility is proportional to G with w being the (real) frequency of

the light (driving force), the real and imaginary parts of G are thus smooth, the poles on the lower half of the frequency plane being sufficiently far away from the real axis. Only in the academic case in which one is interested in every single frequency does one set I:: equal to zero in (23.9) to obtain the real Green function


and to investigate the poles along the real axis. In practical calculations this is feasable only for eigenfrequencies sufficiently far away from the phonon bands, that is, for frequencies of gap or localized modes.

The equations of motion including damping terms yield in the same way eigenfrequencies at the poles of the Green function; in both cases the secular equation is

det(I+Go V)=O (23.11)

with W= ±wA -iYA' G as a function of real and imaginary parts of Go = G~ + i G~ can be written

as

G=G' +iG"= [(I+G~ V)2+G~' VG~ V]-l

. [(I+G~ V) G~+G~' VG~+iG~']

with G~' V=(I+G~ V) G~ V(I+G~ V)-l.

Correspondingly the T matrix becomes

T= T' +iT" = [(I+G~ V)2+G~' VG~ V]-l

. [V(I+G~V)-iVG~'V].

In the linear (Einstein oscillator) case this reduces to

(1 + g~ v) go +g~ vg~ +ig~ g (l+g~v)2+(g~vf

v(l+g~v)-ivg~v

(1 +g~V)2+(g~V)2'

For frequencies outside the bands G~ vanishes, and we have from (23.12)

G"=O

G' = (I + G~ V) - 2 (I + G~ V) G~ = (I + G~ V) - 1 G~.

Here the poles of G' are obtained from

det (I + G~ V) = 0

(23.12)

(23.13)

(23.12')

(23.13')

(23.14)

in close analogy to (22.21). Since G~ is smooth, (22.11) may no longer have solutions for arbitrary V, or it may have solutions within the bands which supply the resonance frequencies. This will be discussed in the next section.

Example: The isotopic mass defect in the NaI lattice. The example of Sect. 22e may serve to illustrate the foregoing development. Figure 22.5 shows a simplified version of the real Green function, calculated from the breathing-shell model for NaI. In Fig. 23.1 is plotted the complex analog of the real Green function. Curve a shows the real part of go(O, IX; 0, IX) with 0 referring to a negative-ion site of NaI. Curve b shows the corresponding imaginary part. The curves are calculated from eigenvalues and eigenvectors of the breathing-shell model with the use of 262 q vectors in the reduced Brillouin zone (a total of 8000 in the first Brillouin zone). The delta functions in the


.,

101

E ~ IJl

'" C >-~ ... 12 Q

Nol: -0.5

Fig. 23.1. Real and imaginary part of the complex Green function versus frequency for a negativeion site in N aI. Curve a: g~; curve b: g~; curve c: -llv; curve d: gO.

imaginary part were calculated as a function of w2 and approximated by a histogram with step width w!ax/100; the histogram was smoothed by straight lines connecting the midpoints of the steps. The real part was calculated by the Hilbert transform of the so-obtained imaginary part.

Even though the real part of the complex Green function does not seem to show any resemblance to the real Green function, note that both curves are rather similar in the frequency regions outside the phonon bands. In addition, the real part of the complex Green function shows strong variation at frequencies where the "residuum" is strong. This occurs where the imaginary part makes large contributions; this behaviour is a consequence of the Kramers-


Kronig relation for the real and imaginary parts, see (34.83), (34.84) and Sect. 24a.

The solutions of (23.l4) can be found graphically from curves a and c in Fig. 23.1, as was done in Fig. 22.5. Notice the following:

·1) Local or gap modes occur only within a certain range of values of ,1 m. In this specific example there is essentially no F15 gap mode for heavy defects (8)0).

2) If ,1m is below a critical value there will be no solutions to (23.l4) at all. 3) There is a small and mostly even number of solutions. 4) Because of the similarity between the real part of the complex Green

function and the real Green function, the solutions outside the bands are about the same. There are solutions to (23.14) in the bands which are called quasilocalized or resonant modes, to which we will turn below. In fact, the behaviour of the perturbed Green function on the defect properties is determined to a large extent by the solutions of (23.14).

c) Resonances: Localized and resonant modes. While the condition

det [I + Go V] = 0, OJ complex

supplies the poles of the eigenfrequencies in the complex frequency plane just below the real frequency axis, there are only a few solutions (if any at all) for OJ

on the real frequency axis as the example shown in Fig. 23.1 (curves a and c) demonstrates. Here a resonance-like behaviour may occur if

Re det(I+Go V)=O, OJ real (23.15)

holds because G or T, see (23.l2), (23.13) are proportional to [det (I + Go V)] -1.

Equation (23.15) is called the resonance condition. A solution to (23.15) is called a resonance, the corresponding frequency the resonance frequency. For local and gap modes, where G~ = 0, (23.15) is identical to (23.l4). The solutions of (23.l5) for frequencies within the phonon bands are said to lead to resonant or quasi-localized modes. It will be shown in Sect. 24c that

1m det(I+Go V) (23.16)

is related to the width of the resonance which becomes obvious in the simple Einstein case exhibited in (23.12') and (23.l3'). From there it is clear that a sharp and strong peak of the imaginary part of g and thus of the susceptibility occurs only if expression (23.l6) is small in which case one has a true resonance. In the case in which expression (23.l6) is large one speaks of an antiresonance since the opposite of a resonance-like behaviour of the Green function occurs. The Einstein case demonstrates this. Here the resonance condition is

1 +g~v=O. (23.l7) Then from (23.l2')

g" = g~/(g~ V)2 = g~2 / g~

at the resonance frequency. For g" to be large, one has to require that g~ be small. As a rule of thumb g~ is small, where the slope of g~ is positive, and g~ is large, where the slope of g~ is negative, cf. Fig. 23.1 (curves a and b). This is a


property of the complex Green function whose real and imaginary parts are connected to one another by Kramers-Kronig transformation, as mentioned above. So it is only in the first case (g~ small, ag~/aco2 >0) that one has a true resonance, while the second case usually leads to a depression of g" whence it is called an antiresonance.

It was mentioned above that there are usually pairs of solutions to (23.12), that is, for every resonance there is an antiresonance. One may say that the intensity of g" from the antiresonance frequency is reshuffled into the resonance frequency region. This sounds like a sum rule. Using the representation (23.9) or (23.10) of the Green function and the normalization of the eigenvectors, (22.6d), one finds

00 00

S coG"(co) dco= S dco2 G"(co)=nM- 1 =n(Mo+LlM)-l. (23.18) -00 0

The integration over both positive and negative frequencies corresponds to an integration over absorption and emission spectra. Since G"(-co)= -G"(co) holds, the intensity of the absorption spectrum is equal to that of the emission spectrum and equal to one half of the total spectrum.

If the defect mass is not involved (as in the even parity vibrations of the example of Sect. 22d), the total area under the curve of each element of G" is in fact a constant (zero for the off-diagonal elements), independent of the perturbations.

The area under the curve of only the defect-site element is affected by the defect, while that of the off-diagonal elements is zero.

Contrary to a local or gap mode which is a well-defined eigenvibration of the crystal there is actually no "resonant mode" of the same type, the latter is rather a large number of vibrations which have frequencies close to the resonance frequency and which participate in a large-amplitude vibration.

There are situations in which Re det(l + Go V) in some frequency regions does not have a zero but an extremum close to a zero. In this case, there is apparently no resonance according to the condition (23.15), but the imaginary part of G will show a strong enhancement due to a relatively small denominator. This is often referred to as an incipient resonance.

Example: The isotopic mass defect in the NaI lattice. Consider again the case of NaI with an isotopic negative defect. The real and

imaginary parts of go(O,!X; O,!X) have been plotted in Fig. 23.1. For a relative mass change of I!.=Llm/m= -0.75, three resonances and antiresonances are encountered as solutions of 1 + g~ v = 0 represented by the intersections of the curves a (g~) and c ( -1/v = l/co2 LI m) in Fig. 23.1. The imaginary part g" of the perturbed Green function corresponding to this perturbation is shown as curve d in Fig. 23.1. Upon comparison of g" with g~ (curve b), the enhancements at the resonance frequencies as well as the depressions at the antiresonances are clearly seen. At higher frequencies, g" shows appreciable scatter. This is mainly due to numerical uncertainty.

Given a mass change, LI m, (23.17) may not always have a solution, e.g. for very small values of ILlml. On the other hand, given a frequency, one can

322

Acoustical band -1

... r\" -1

0 ----------

(


Gap Optical band :0

.10 .10

Z

------- Nal:+-------

14>14(

II)

--------Nal:- ------

Sect. 23

Fig.23.2. Relative mass change, e=,1m/m, versus resonance frequency, w, for substitutional isotopic defects in N aI. Full portions of the curves: resonances; dashed portions of the curves: antiresonances; dotted portions of the curves: unphysical solutions (negative defect mass required to

produce resonances). The ordinate of the curves marked x 10 is reduced by a factor of 10

compute the mass change necessary to produce a solution of (23.17). The results of such a calculation are shown in Fig. 23.2. The upper part plots the fractional mass change 8 of the defect substituting for a light (Na +) ion of the NaI host lattice versus frequency; the lower part shows the same for the defect substituting for a heavy (1-) host ion. Oscillations in the optical-frequency region have been drastically smoothed out in the drawing.

The following statements generally are true and are illustrated by Fig. 23.2: 1) Upon substitution of the light ion, a local mode rises out of the optical

band when the mass of the defect is below a certain critical value. If a heavy defect for the light ion is substituted, a gap mode drops out of the optical band. If a light defect is substituted for the heavy atom, a local mode rises out of the optical band and a gap mode out of the acoustical band. A heavy defect replacing a heavy host ion causes neither a local nor a gap mode (MAZUR et aI., 1956).

2) As mentioned earlier, the range of values of .d m which produce a local or a gap mode, is restricted.

3) With increasing mass, the local and gap modes move into the optical or acoustical bands respectively, and turn into resonant modes.

4) In regions with high density of states there are anti-resonances (dashed portions of the curves) rather than resonances.


5) There are unphysical solutions (dotted portions) where the defect would have to have negative mass (8 < -1) in order to generate a resonance.

Calculations illustrating points 1) and 2) above for NaI have also been made by JASWAL (1965 a). PLUMELLE et al. (1979) have done similar calculations using a rigid-ion model for the cuprous halides and incorporating force-constant changes to nearest neighbours.

If changes in force constants are involved, and if the linear case is a sufficient approximation, -1/v = l/w2 .1 m, curve c in Fig. 23.1, is replaced by -1/v = - (.1 f - w 2 .1 m). The pole of the corresponding curve is shifted from w = 0 to higher frequencies if .1 m . L1f is positive. The resonance frequencies shift to higher frequencies with increasing L1f as expected, while the antiresonance frequencies shift to lower frequencies.

d) Eigenvalue treatment of the Green function and T matrix in the impurity space. In order to be able to work with linear quantities (or with vectors at most), it has been suggested that one should work with eigenvalues and eigenvectors of go v (MONTROLL and POTTS, 1955; KLEIN, 1963, 1966, 1968b; MARADUDIN, 1965). The discussion of the properties of the Green function, as in Sect. 24, will be much facilitated.

Let l/Ij denote the eigenvectors of go v and let Pj be the corresponding eigenvalues

(23.19)

where Pj is a complex quantity as is go v. The left-hand eigenvectors of go v must be different from l/Ij since go v is not symmetrical. One easily sees that the lefthand eigenvectors are l/IJ v,

One also has

and go can be represented as

and finally g as

(23.20)

whereas t turns out to be

(23.21)

The factor 1 1

I+Pj I+Pj+ipj (23.22)

has poles below the real frequency axis; for real frequencies a resonance with frequency wr occurs where

(23.23)


holds for some j equal to r which is equivalent to

Re det(1 + go v) =0.

Sect. 24

(23.24)

As g~ and g~ are Kramers-Kronig transforms of each other so will g~ v and g~ v be except possibly for (generally infrared-active) vibrations in which v contains the term w 2 L1 m and therefore depends on frequency. In all other cases one expects the same to hold for J-tj and J-tj'. This means that one can repeat the arguments of Sect. 23d on resonances and anti-resonances. Since v is a monotonic function of frequency (if it depends on it at all), the qualitative arguments are unchanged for the infrared-active vibrations, too.

For a resonance (w = wr ) only the r-th term with 1 + J-t~(wr) = 0 survives in (23.20) and (23.21) unless J-t:(wr) is large, which is the antiresonance case. Also, if

(1 + g~ v) "'r=(l + J-t~) "'r=O

for a local- or gap-mode frequency then "'r must be - except for a normalization factor - the same as the eigenvector ;{(r) for that particular mode; in all other cases this is true only to zeroth order in J-t: (KLEIN, 1968b).

In the following section various conclusions from the frequency dependence of J-tj will be drawn. In the linear case one can equivalently substitute go v wherever J-tj appears.

24. Properties of the perturbed harmonic lattice Green function

a) Kramers-Kronig transform. In (34.83) and (34.84) of Chap. G it is shown that the real and imaginary parts of the Green function are Kramers-Kronig transforms of one another. This is very important for actual calculations which involve the complex Green functions, because, after computation of the imaginary part of G, the real part can be obtained by Kramers-Kronig transformation from the imaginary part,

Of G" we need to consider the impurity-space submatrix G;; only, since the other parts can easily be obtained from G;; according to (23.8).

Part of the literature prefers to use

G1 =Xo[W&-(w2+is)I]-1 X6

instead of the retarded Green function

Go = Xo [w~ - (w +is)21] -1 X6. In (34.82) it is shown that

and one can easily see that

Sect. 24 Properties of the perturbed harmonic lattice Green function 325

About the use of Go rather than GI see the discussion by WAGNER (1967a). Within one half of the frequency axis, Go as well as GI are functions of w 2 rather than of w.

b) Normalization of the perturbed resonance-mode eigenvectors: An effective mass of the resonance vibration. The contribution LlG to the Green function from a local or gap mode with frequency Wr is of the form

LlG- t/frt/fJ 1 - t/fJ Mt/fr w; -(w+ie)2·

(24.1)

The eigenvector t/fr rather than x(r) is used to indicate that the former way may not be normalized according to (22.6d). In practical calculations one obtains the real part of the Green function from the imaginary part by a Kramers-Kronigtransformation. The imaginary part, however, has a vanishing numerator outside the unperturbed phonon bands, see (23.12), since in that region G~ is equal to zero. Here the real part can be obtained directly from (23.12), but for the calculation of the imaginary part one has to use (24.1); Wr and the eigenvector t/f r can be obtained from the eigenvalue equation (22.16),

(1 + GO(wr) V)t/fr=O,

and only the normalization factor needs to be determined. The normalization condition (22.6d) reads

since, for arbitrary w,

hence

1 = t/fJ Mt/fr = t/fJ(LlM +Mo) t/fr =t/fJ[LlM + Ga l (oG% w2 ) Gal] t/fr

oG/ow2 = GMG.

F or a resonance outside the bands one has from (24.2)

Gal t/fr = - Vt/f"

(24.2)

(24.4)

(24.5)

which will be used repeatedly in Sects. 24 and 25, and the normalization condition becomes

Since LlM and V are restricted to the impurity space, this becomes

with an effective mass (see also KLEIN, 1968b; PAGE, 1974)

m*=Llm+m~

m~= -ogal/ow2=gal(og%w2)ga~

(24.6)

(24.7 a)

(24.7b)


which is in analogy to (24.3). m* is restricted to the impurity space, and the index r in (24.6) indicates that it is evaluated at the resonance frequency wr' With

t/lJI mit/lrI = t/lJMt/lr = L M(L) It/I(L, o:lrW La

it is easy to see that the effective defect mass is bigger than the actual defect mass (PAGE, 1974),

(24.8) a La

The equality holds if the eigenvector is non-zero only at the defect site. Note that mi is not diagonal, that it depends on the size of the defect space, and that it depends on the frequency of the resonance vibration. In addition, t/lr and t/lrI depend on the actual perturbations.

The interpretation of the effective mass is the following. Due to the fact that the lattice takes part in a resonance vibration, the inertia of the lattice lowers the resonance frequency compared with the one obtained from an Einstein oscillator. This effect is expressed as an enhancement of the mass. Since the lattice takes part in resonance vibrations to different degrees at different frequencies, the effective mass is a function of frequency.

For resonant modes things are slightly more complicated, since the poles of the Green function are located off the real frequency axis. Only for these frequencies does (24.2) hold. If, however, the imaginary part of Go is very small at the (real) resonant-mode frequency determined from (23.14),

det(1 + G~ V)=O, then

(1 + G~ V) t/lr=O

is a good approximation (KLEIN, 1968b), and the arguments above can be repeated with the real part of the Green function substituted for the real Green function.

c) Approximate form of the Green function and of the T matrix near a resonance frequency: Width and intensity. The T matrix is given in (23.21) as

t = L vt/lj[(l +,u) t/lJ vt/lj] -1 t/I; v. j

Near a resonance frequency one has from (23.23)

l+,u:(wr)=O.

(24.9)

(24.10)

For w~wr only the term withj=r in (24.9) is important, and one expands the denominator in (24.9) as

1 + ,ur(W) = 1 + ,u~(wr)+(w2 -w;) a,u~/aw2 +i,u~(wr) (24.11)

assuming higher-order terms to be negligible. Using (24.11) and (23.19),

,urt/lr=gO(wr) vt/lr'

Sect. 24 Properties of the perturbed harmonic lattice Green function

one has for w ~ Wr

t = v l/I r {[(w2 - w;) 0 1l~/OW2 + ill~] l/I: v l/I r} -1 l/I: v

= v l/I r [(w2 - W;) l/I: v(o g~ v/O(2) l/Ir + i l/I: v g~ v l/I r] -1 l/I: v

v l/Ir l/Irt V 1 l/Irt m~l/Ir w; -w2 - i2wr Yr

with the effective mass m~ from (24.7a) and a width Yr ,

2wr Yr=l/I: v g~ vl/lrll/l: m: l/Ir = l/I: (g~) -1 g~ (g~) -1 l/I rll/I: m~ l/I r'

Correspondingly, the Green function for W ~ wr is

G=Go-GotGo,

in particular, the defect-space Green function becomes

l/Irl/l: 1 g=gO+,I,t *,1, 2 2'2

'I'rmr'l'rWr-W -1 WrYr

327

(24.12)

(24.13)

(24.14)

if g~ v l/Ir is small. Expression (24.14) reduces to (24.1) under the restrictions placed there.

The Green function as given by (24.14) indeed displays a resonance behaviour near a resonance frequency. The intensity of the resonances are determined mainly by the slope of g~, i.e. by the effective mass, see (24.7). The width is determined mainly by the imaginary part of go; it is zero outside the bands in which case delta functions are obtained for glf.

In general, Yr is a function of frequency and therefore leads to a distorted Lorentz curve in the region of the resonance. This is very similar to the effects of the frequency dependence of the anharmonic self-energy, see Sects. 30e and 30i. Inspection of Fig. 23.1 shows that g~ is small only where og~/OW2 is positive. The considerations above are therefore valid only for resonances and not for antiresonances; one is generally much less interested in the latter, anyway.

Example,' The isotopic mass defect in the N aI lattice For the resonance of an isotopic defect one has

Il=go V= -go w2 L1m and

(24.15)

For the three resonances found from Fig.23.1 one can make a rough estimate of the intensity and width of gil by determining the slope of g~, i.e. of 11', graphically. The Lorentz curves thus found are shown in Fig. 24.1 as dashed curves together with the exact curve (full line). The intensity of the gap-mode line was found to be about that of the resonance line in the optical band frequency region. It is seen that the optical resonance line is well approximated by the Lorentz curve; the frequency is slightly off. In the acoustical region the peak position and height are correct, but the width, and thus intensity, is far too large. This apparently is the result of the strong increase of 1l"(W2), i.e. of g~, in

2.5

2.0

1.5

1.0

0.5

o


0.5 1.0 1.5 2.0 3.0

Sect. 25

3.5 .1013 sec-1

Fig. 24.1. Intensity and width of resonances. The full curve gives the exact imaginary part of the perturbed Green function, see Fig. 23.1. The dotted curves are Lorentz approximations, see text

the wings of the resonance line (cf. Fig. 23.1) which leads to an increasing width with increasing distance from the resonance frequency.

25. Applications of Green functions: Phonon spectra in perturbed crystals. The Green functions cannot be observed as such. They appear implicitly in various crystal properties, of which three examples will be discussed. Two, the infrared and Raman one-phonon spectra (Sects. 25b and c), are the subject of the present article, the other one, the phonon density of states (Sect. 25 a), is very simple and therefore instructive.

Even though the applications below are not restricted to harmonic Green functions, it is felt that this is the appropriate place to demonstrate the effect of harmonic perturbations (rather than the effect of anharmonicities) not only on phonons but also on physical properties. The consideration of anharmonic effects is deferred to Sects. 30 and 31, where the present results will be generalized. Generalizations to a model with deformable ions and to finite concentrations will be found in Sects. 27 and 28.

a) Phonon density of states. Let N be the number of elementary cells in the crystal and let s be the number of atoms within each cell. Then the total num-

Sect. 25 Applications of Green functions: Phonon spectra in perturbed crystals 329

ber of vibrations in a three-dimensional crystal is 3sN. The number of states within a frequency interval Wo ± L1 w is given by

L c5(w;.-w) ;.

wo-L1w<w;.~wo+L1w.

For a vanishingly small interval this becomes

L c5(w;. -wo)· ;.

If instead one asks for the number of states on an w2 scale rather than on an w scale one has

This quantity is closely related to the imaginary part of the Green function,

G" = L I(A.) It (A.) n c5(w~ - w 2 ) ;.

for w > O. Using the normalization condition (22.6 d)

I t (A.) M I (A.) = 1, one finds

1 =- L It (A.) M I (A.) n c5(w~ _w2 )

n ;. 1

=- Tr[M L I(A.) It (A.) nc5(w~-w2)J n ;.

=~Tr(MG"). n

Let the density of states, p(w2 ), be normalized to 1. One then has

1 1 p(w2)= 3sN ~ c5(w~ _w2)= 3sNn 1m Tr{MG}.

The unperturbed density of states is analogously given by

2 1 { Po(w )=3sNn ImTr MoGo}· (25.1)

The difference between p(w2 ) and Po(w2 ) is found by using M=Mo+L1M and (23.1),

Tr(MG)=Tr[(Mo+L1M)(Go- GoTGo)]·

Writing

and using (24.4),

330

one has


Tr(MG) = Tr(MoGo)+ Tr(LI MG)- Tr(MoGo TGo)

=Tr(MoGo) + Tr(LI mg) - Tr(t8 go/8w 2),

Sect. 25

since LI M and T are nonzero only in the impurity space. This can be written either as

Tr(M G) =Tr(MoGo) - Tr[(l + go V)-l 8go v/8w2] (25.2)

or, taking the effective mass from (24.7), as

Tr(MG)=Tr(MoGo)- Tr(m~go)+ Tr(m* g).

Then one has

p(W2)_PO(W2)=3s~n ImTr(m*g-m~go)·

With (23.l9) one obtains from (25.2)

p(w2)= Po(w2)_~l~ 1m Tr (I 8/l/8w2 "'~"'; V) 3sNn j l+/lj "'j V"'j

_ (2) 1 I '\' 8/l/8w2 -Po w -~~ mL....

3sNn j l+/lj

1 =Po(w 2)--3 N ImI8ln(l+/l)/8w2 .

s n j

(25.3)

(25.4)

The perturbed density is different from the unperturbed one by terms of the order of liN. This is expected as long as one considers only one defect per crystal.

The density, p(w2 ), may show a resonance behaviour as discussed in Sect. 24c. Special care has to be used at local- and gap-mode frequencies. From (23.23) a resonance occurs when

for a particular r. For w~wr one then has as in Sect. 24c

and

with

1 + /l~(W)~(W2 -w;) 8/l~/8w2,

/l~/(W)~ /l~(wr)

(25.5)

As one expects from Sect. 24b, (25.5) reduces to b(W2_W;)/3sN forlocal and gap modes where )ir --+0.


MARADUDIN (1965) presents an alternative method which starts from the following representation of the unperturbed density of states,

2 1 -1 Po(W )=3sNn ImTr[Mo(<Po-zMo) ]

1 d = -3sNn ImTr dz In(<Po-zMo)

1 d = --- 1m -d Indet(<Po-zMo)·

3sNn z

Using this method one can easily show (LITZMAN, 1966) that

1 d p(w 2 )=PO(W2)--- ImTr-d In(l+Go V)

3sNn z

the first equation being in accordance with the result (25.4). In fact, this representation goes back to KREIN (1953).

b) Dielectric susceptibility. The absorption of light by defective crystals was first formulated in terms of perturbed Green functions by LIFSHITZ (1943a, b, 1944). Lifshitz' result did not become known outside Russia until 1956 when he published a review in a non-Russian journal. While his theoretical work slept like the Sleeping Beauty, interest in defect-induced absorption was created by the infrared spectra of type I and II diamond (see, e.g., ANGRESS et aI., 1965b). LAX and BURSTEIN (1955) suggested that the spectra showed defect-activated critical points. SZIGETI (1963) made a linear-chain model calculation, as was previously done by WALLIS and MARADUDIN (1960) to support this idea. A three-dimensional Green function calculation of the absorption caused by charged defects in (neutral) group IV and rare-gas crystals was first presented by DAWBER and ELLIOTT (1963a, b), ELLIOTT and DAWBER (1965) and for homo polar crystals by GUREVICH et al. (1964). It was found that the vibrational dipole moment through which the radiation is absorbed is caused by the sole motion of the defect, which carries the only charge. For rare-gas crystals see, for example, COHEN and KLEIN (1974) and the references given there. JASWAL and WADEHRA (1975) suggest a formulation closer to standard quantum mechanics.

In polar crystals every ion is charged and radiation is absorbed by the crystal as a whole. If the defect carries the same charge as the substituted host ion, perturbed phonons are excited only via virtual excitation of a Reststrahlen phonon. This was shown in a linear-chain model calculation by WALLIS and MARADUDIN (1960). The general expression for three-dimensional crystals and especially alkali halides was first given by MARADUDIN (1963). TAKENO (1967, 1968a) and PATNAIK and MAHANTI (1967) gave theoretical results for a very simple three-dimensional model (with short-range forces only). Numerical results for defects in alkali halides have been presented by GUNTHER (1965b);


BENEDEK and NARDELLI (1967b); and others. Martin has presented results for monatomic f.c.c. crystals (MARTIN, 1967a) and crystals with the CsCI structure (MARTIN, 1971; see also AGRAWAL and RAM, 1971; RAM and AGRAWAL, 1972a, b, c; who in part arrive at literally the same conclusions). The CaF2 structure has been treated by HAYES and WILTSHIRE (1973); HAYES et al. (1975); MnF2:Tm by BENSON (1973); and ZnP2:As by ARTAMONOV et al. (1979); just to give a few references. An extensive study using a shell-model representation (see Sect. 27c) has been given by MACPHERSON and TIMUSK (1970b) and HAQUE and STRAUCH (1977; HAQUE et al. 1977, 1978).

As mentioned above, absorption by defects in host crystals with neutral constituents has to be via charge perturbations. Rare-gas defects in rare-gas crystals with zero charge require non-zero polarizability changes in order to cause absorption (HARTMANN and ELLIOTT, 1967; BELLOMONTE and PRYCE, 1968). While the previously mentioned considerations are applicable to rigid-ion models, polarizabilities require the use of shell models. The expression for the absorption constant is readily generalized to shell models (STRAUCH, 1968; MACPHERSON and TIMUSK, 1970b) and will be presented in Sect. 27c.

On the other hand, LEIGH and SZIGETI (1967 a, b, 1968) argue that the distortions around a defect induce charge perturbations at the neighbors of the defect. They estimate the magnitude of these "apparent" charges from photoelastic constants. In addition to this effect there may be charge perturbations from quantum-mechanical charge transfer. The effect of such perturbations on the absorption has been investigated by MARTIN (1968) whose results will be discussed below. Recently, GRIMM (1972) evaluated the charge tensor of impurities in homopolar crystals.

References for the absorption due to defect-pair vibrations can be found in Sect. 20i and 29c. The off-center Ag+ defect in RbBr has been treated by MOKROSS and DICK (1977); see also MOKROSS et al. (1977).

The absorption constant, K(w), is defined in terms of the relative transmitted light intensity, 111o, and is given by

K(w) = (1/d) In (IoII)

= (4nwlnc) ;(" (w)

without corrections for reflection losses and interference effects. Here d is the crystal thickness, w is the incident light frequency, c is the velocity of light, and n is the refractive index, which is generally assumed to be a constant. Finally, X"(w) is the imaginary part of the dielectric susceptibility, which is in general a tensor.

Even though it is generally assumed, n is not a constant; in fact, where X"(w) shows strong variation, n varies, too. In pure crystals this occurs in the Reststrahlen-band frequency region. In perturbed crystals, the frequency region of interest is that of low host absorption. In that region the variation of the purecrystal refractive index is small. The defect-induced changes in n are indeed expected to vary strongly in the regions of resonance frequencies. However, to first order, these changes will be proportional to the defect concentration, which is generally very small. It will be a good approximation to take n to be that of the host crystal and to be constant.


In the harmonic approximation the dielectric susceptibility tensor X~p(co) is obtained from (8.24) and (34.50) as

with 3

1 " 2 co;. X~P(co)=hV L.M~(A.) 2 ( +')2 Mp(Je) ;. CO;. - CO Ie

1 = V L Z~(L, y) G(L, y; E, y') Zp(E, y')

LL'yy'

1 T =VZ~GZp

M~(Je)= L Z~(L, y) X(L, yl Je) (h/2co;.)1/2. Ly

The charge of the defect and of its neighborhood may be perturbed,

Z~=Zo,~+AZ~.

The perturbed Green function is from (23.1)

G=Go-GoTGo,

and one finally obtains

1 - - T X~p(co)=V (Zo,~+AZ~) (Go-Go TGo)(Zo,p +AZo,p)' (25.6)

The use of the completeness relation (22.6) enables one to establish a sum rule for the total integrated absorption as 4

00 1 00 2n 00

J nK(co) dco =- J nK(co) dco =- J X" (co) dco2 o 2 -00 c 0

(25.7)

with 00 1 00

J X~p(co) dco2 = V Z~ J G"(co) dco2 Zp o 0

=; Z~ ~ Z(Je) zt(Je) Zp

=!:. ZT M- 1 Z V ~ p

(25.7')

In the rigid-ion approximation the total integrated absorption thus depends only on the charge and mass changes introduced by the defects, a light (heavy) impurity generally leading to an absorption increase (decrease) if charge per-

3 We hope, the reader will not be confused because the same symbol, X, denotes the susceptibility and the eigenvector.

4 See the comment following (23.18). Where the refractive index varies appreciably with frequency, the integration has to be performed as indicated.


turbations are neglected. In the shell models the effective charge Z depends on force constants of the models as will be shown in Sect. 27 c. Whether introduction of defects into a crystal will enhance or diminish the total absorption will, therefore, depend on the force-constant changes as well.

Expression (25.6) for the susceptibility can be simplified. First, one uses the fact, that Ll Za has components in the impurity space only. This results, with Ll Z~ = (Ll z~, 0) as in Sect. 22e, in

TG Ll Z = (t go Ll zp 0) (25.8) o p 0 O·

Second, one uses the lattice translational invariance of the unperturbed quantities Go and ZO,a' With (22.4),

Xo(l, K, a I q,j)=(Mo, KN)-1/2 e(K, a I q,j) exp iq· xCi, K)

the Green function, (22.15), can be written as

Go(l, K, a; 1', K', al)=~ L <:9'O(K, a; K', a'l q) exp iq· [xCi, K)-X(l', K')]. (25.9) N q

The product Go Zop requires

L Xo(l, K, a I q,j)Zo,p(K, a)= L (N /M 0,K)1/2 e(K, a I O,j) Zo,p(K, a) (25.10) lKa K.a

where one has used

L exp iq . xCi) =N Ll (q). (25.11) 1

Thus one finds (GOZo,p)lKa = L <:9' o(K, a; K' a'l O)Zo,p(K', a') (25.12)

K'a'

where there is no explicit dependence on xCi) any more. Had one kept the nonvanishing but small wave vector of the light it would be this which would appear in (25.12) instead of q = O. Also one would have found that only transverse vibrations would couple through the charges Zo to the light what is physically clear.

In simple cases the eigenvectors for q = 0 can be determined from symmetry considerations. In particular in diatomic crystals the optical-mode eigenvectors are determined from the condition that the center of mass be at rest and from the normalization condition (22.6b), and they turn out to be (except for a common phase factor)

e( +, a I TO) = (,u/M +)1/2,

-e( -, a I TO) = (,u/M _)1/2, (25.13 a)

here + and - refer to positive and negative ions, ,u is the reduced mass, and TO refers to a transverse optic vibration at q = 0 and polarized along the a direction. With diagonal charge tensors,

Zo,p(+,a)=ZObap ,

-Zo,p( -, a)=Zo bap , (25.13b)


one finds that for the diatomic crystal (25.10) becomes

L (NIM o,K)1/2 e(K, IX I TO)Zo,p(K, IX) = (NI/l)1/2 Zo (25.10')

for a TO mode polarized along the fJ direction. The unperturbed contribution to (25.6) becomes

T Z~ 1 Zo"GoZop=N- 2 2' , , /l coTO-co

(25.14)

while the perturbed quantities involve either expressions like (25.8) or TGoZo; in this case one of the eigenvectors appearing in Go, (25.9), is treated as in expression (25.10') while the other as a defect-space vector like in Sect. 22d, for example. The notation shall be

ZL Go Zo,p=N zI;,,, <;l}0(0) zo,p

t Go Zo,p =t <;l}0(0) zo,p

(25.15a)

(25.15b)

for two- and multi-atomic crystals. The susceptibility X"p(co) can now be written as

V T 1 T T N X"p(CO)=zo, ,,<;l}0(0) zo,p+ N {-zo,,,<;l}o(O) t<;l}o(O) zo,p+Llz" gLlzp

+ zI;,,, <;l}0(0)(1- tgo) Ll zp + Ll z~(I- go t) <;l}0(0) zo, p}. (25.16)

The susceptibility is given by that of the unperturbed crystal (first term) plus a defect-induced part (in curly brackets). If one adds terms of the order (1/N)2 one can group various terms together and ends up with two terms,

with

V N X"p(co)=z~(<;l}o(O)_N-l <;l}0(0) t <;l}o (0)) zp

+N- 1 Llz~ gLlzp+O(N- 2)

z,,= zo,,, + N- 1 (1- tgo) Ll z"

=zo, ,,+ N- 1 (1 +vgO)-1 Llz" +N -1 -1 A =zo,,, go gLJZ".

(25.17)

(25.18)

The first term of (25.17) gives a modified harmonic Reststrahlen susceptibility, the modification of the propagator being brought out more clearly by the low concentration result of Sect. 28 e. This term describes the excitation of the pure-crystal Reststrahlen (TO) phonons which are or are not scattered at the defect; scattering occurs with probability liN which is the fraction of sublattice sites occupied by defects.

The (effective) ionic charge, (25.18) by which the light couples to the Reststrahlen phonon, is also changed by a small fraction due to charge perturbations (charge transfer, change of polarizability, etc.) at the defect. In this connection we refer to the shell-model treatment of perturbed lattice vibrations in Sect. 27. Note that the charge z is a complex function of frequency.


The second term of (25.17) describes direct excitation of perturbed phonons. It is to be noted that here the perturbed phonons are excited, not via the total charge of the defect but only via the change in charge.

In the case in which there are no (or negligible) charge perturbations, the second term of (25.17) is zero, and one is left with

~ XafJ(W)=Z6,a(~0(0)-N-1 ~0(0) t~o(O)) zO,fJ'

Writing the t matrix as in (23.6'),

t=v-vgv,

one can express the defect-induced part alternatively in terms of the perturbed Green function.

In Sect. 28 it will be shown that expression (25.17) is the low-concentration approximation for successive scattering at different defects. Restricting the discussion to the linear case for simplicity and to the impurity-space contributions, the terms in brackets can be written to order liN as

~0(0)_N-1 ~0(0) t~o(O)= [(~0(0))-1 +N-1 t]-l +O(N-2). (25.19)

In Sect. 28 the term tiN is interpreted as a harmonic self-energy, and in Sect. 30e it is shown that the imaginary part of an expression like (25.19) near a TO phonon frequency can be approximated by a Lorentz curve in a way very similar to that in Sect. 24c. The area under this curve is then proportional to

(1-N- 1 8t'18w2)-1

evaluated at a frequency Wo (near the corresponding Reststrahlen frequency) which is obtained from

wio-w~+N-1 t'(wo)=O.

Whether the Reststrahlen intensity is enhanced or decreased depends on the sign of the slope of t', which in turn depends on the system under consideration.

At frequencies other than the Reststrahlen frequencies WTO' the absorption constant is proportional to

~ X~fJ(w)= _N- 1 Z6,a~~(0)t"~~(0)zO,fJ' (25.20)

The defect-induced spectrum is seen to depend via the scattering matrix on the perturbations introduced by the defect. This is demonstrated in Fig. 25.1, taken from the paper by MACDONALD et al. (1969) which was mentioned above in Sect. 21. These authors have shown that the theoretical spectra depend very strongly on the details of the perturbations introduced by the impurity. In order to reproduce the resonance-mode peak in the spectrum induced by F - defects in NaCl, they had to allow for a short-range central force-constant change, LJf1' between the defect and its nearest neighbors of about 90 % (dashed curve in Fig. 25.1 a). This force-constant softening is surprisingly large and probably beyond the value at which an instability of the defect occurs. In any case, strong perturbations between the defect and its nearest neighbors are expected to cause


ANGULAR FREQUENCY (IQi3rad/sec) ANGULAR FREQUENCY (lo'3rod/sec) 1.0 1.5 2.0 2.5 3.0 1.0 L5 2.0 2.5

30r----r-----r-----r----~----_,--~ 5000,----,-----,-----,------,----

25

I- 20 ;z

~ (f) ;z

815

;z Q I-

~ 10 o (f)

ID ct

5

NoCI.O.IMOLE % F-

TO' TO --- CENTRAL FORCE 354--/ 400 MODEL eoi': eoi' - LATTICE RELAXATION

I MODEL I I I

J I

'e ...

200.0

100.0

50.0

;: 20.0 z ;'$ ~ 10.0 o t.)

~ 5.0

NoCI+O.1 MOLE % F-

7°K

••• EXPERIMENT

-THEORY

0L-~~£'-----------L----------~--~

llL It: g CD 2,0 C

• a

50 100 150 WAVENUMBER (em-I)

1.0

0.5

0.2

Fig. 25.1. (a) Calculated absorption of NaCI: F-, with z* = e. Full curve: model with change in nearest-neighbor force constant, -14045 erg/cm2 ,

and change in nearest neighbor/fourth-neighbor force constant, -8000erg/cm2 ; dashed curve: model with change in nearest-neighbor force constant, -14500 erg/cm 2 ; (b) comparison of theory and experiment for NaCI: F- using the model with parameters as for full curve of Fig. 25.1 a. Ul~~~----------~----------~

50 100 150 (MACDONALD et aI., 1969) b WAVENUMBER (em')

distortions and hence further perturbations, as explained in Sect. 22b. In fact, a 50 % decrease in the force constants between the nearest neighbors and fourthnearest neighbors, iJ!2 (i.e., along the cubic axes, cf. Fig. 22.1 b) changed the structure in the spectrum around 125 cm -1 such as to substantially improve the agreement with the experimental curve (solid curve in Fig. 25.1 a), as a comparison with the experimental curve (Fig. 25.1 b) shows. It is interesting to note that the change iJ!2 apparently only weakly influences the resonance frequency near 60 cm - 1, indicating a strong localization of this mode. The vibrations around 125 cm -1 are far less localized, since they do react strongly to the change iJ!2'

At a resonance frequency Wr one has from (24.12) and (25.20)

V "() 1 T <§'(O) 2wrYr V "'r"'; V <§'(O) N Xapw=Nzo.a 0 (2_ 2)2+(2 )2.I,T *.1, ° zo,p (25.21)

wr W wrYr 'l'r mr 'l'r

with an integrated resonance-line intensity obtained from (25.7'),

V Ws "( )d 2 nT' (0) V"'r"';V '(0) O( / ) N XaP W W = N ZO.a <§O .I,T *.1, <§O ZO,p+ Yr W r , o 'Yrmr'l'r

(25.21')


if the resonance frequency Wr is sufficiently far away from any of the Reststrahlen frequencies W TO , when one can neglect the frequency dependence of the Green functions ~~(O). Also, ~~(O), m;, and Yr have to be evaluated at the resonance frequency.

When there are non-vanishing charge perturbations Liz, the light excites the Reststrahlen phonon either via the ideal lattice charge or by the charge perturbation of the defect. In the latter case, the Reststrahlen phonon is excited either directly or after excitation and scattering of phonons at the defect. In this case the expressions for the imaginary part of the dielectric susceptibility are rather involved. (At the end of this section the special case of homopolar crystals will be considered for which zo=O but where there has to be a non-zero charge change in order that there be defect-induced absorption.)

Absorption due to impurities with altered charges has been calculated by GENZEL et al. (1965) for an NaCI linear chain and by MARTIN (1968) for a realistic model of NaCl. Martin investigates the defect-induced one-phonon absorption caused by a heavy defect substituting for a CI- ion. He considers two kinds of charge changes: 1) Only the defect carries an altered charge (e.g. a divalent defect). 2) The charge alteration due to the defect ("deformation charge") is caused by charge transfer from the defect to those two of its six neighbors which lie in the direction of the incident light polarization. Even though it is not quite clear why the other four neighboring ions should not participate in the charge transfer, it is interesting to follow the dependence of the absorption spectrum on the charge increase from z = -1, Liz = 0 to z = 0, Liz = 1. While the absolute value of the charge decreases, the absorption in the resonance region (around 1.25·1013 S- 1) first decreases, upper part of Fig. 25.2, but surprisingly then increases again, lower part of Fig. 25.2. This is due to the various counteracting terms appearing in the expression for the dielectric susceptibility. With increasing ILizl the effective charge z decreases, thus lowering the contribution from the first term of (25.17). But at the same time the contribution from the second term is increased.

While Martin includes the effect of charge changes on the dipole moment responsible for the infrared absorption, he does not include the effect of the same change on the lattice dynamics. However, changes in charge induce changes in Coulomb force constants. Even though it has been shown by PAGE and STRAUCH (1967, 1968; see also Sects. 27c and 30b) that the latter effect is small for localized modes, preliminary calculations by STRAUCH (unpublished) have shown that these might become important for band vibrations. In fact, the low-frequency resonances are different from high-frequency localized modes in that the lattice vibrates to an appreciable extent in the former case (PAGE, 1974; PAGE and HELLIWELL, 1975), while the negative result of charge changes in the local-mode case arises, because the defect in this latter case vibrates in essentially static surroundings.

Finally, in the case of the homopolar group-IV crystals, for which expressions (25.15) vanish identically, one is left with

(25.22)

Sect. 25 Applications of Green functions: Phonon spectra in perturbed crystals

I.O'r---.-----...-----.---:1"--_,

us

;; 0.6

!:!. 3

;£ 0.4

0.2

I.U~--.__--,...--~--___,_--_,

0.8

__ 0.6

':::; !:!. '3 ;£ 0.4

0.2

0.5

I'll = l.Oe

339

Fig. 25.2. Absorption coefficient K(w) for a deformation charge defect with mass 100 at a negativeion site in NaCI. (MARTIN, 1968)

where the susceptibility directly reflects the perturbed Green function at the impurity site (DAWBER and ELLIOTT, 1963b). The Green function, however, reflects the projected rather than the actual density of states. Roughly speaking, the maximum in the density of states at the Raman oscillator frequency is multiplied by matrix elements which depend upon frequency, resulting in zero contributions to the susceptibility from the Raman oscillator. The critical point seen in the absorption spectrum of imperfect diamond a few percent below the Raman frequency is definitely not due to the Raman oscillator but to another infrared-active modes, as correctly interpreted by LAX and BURSTEIN (1955). Apparently, any absorption must be accompanied by a change in charge. This seems obvious. A group-IV impurity in group-V semi-conductors may also lead to some, though probably weak absorption, since the polarizability of the defect is different from that of the host ions. In shell-model terms, this generally means that core and shell charges are changed even though the total ionic charge remains zero. This aspect will be pursued further in Sect. 27.

If resonances occur, the absorption intensities are proportional to

v ooJ "( )d 2_~L1z~l/Irl/l;L1zp N 0 XaP W W - N l/I; m* l/Ir '

5 This is true, if g" ~ g~ holds in this frequency region.


cf. (25.7). In the linear case one has

V Ws "( )d 2_~ Llz~Llzp N 0 X~p w W - N m * .

Here the effective mass determines the strength of the resonance. Another very compact formulation of the dielectric susceptibility can be

given in terms of the eigenvalue representation of go v. This is suggested by (25.21). One writes (25.18) as

z~ = zb.a+ Llz~(l + go V)-1 IN

( 1 1 )t/I.t/lT V t/lTv = ~ zO,a+ N 1 + J.l.j Llza t/lY vJt/lj = ~ Zaj (t/lT vJt/lY/2 (25.23)

and similarly for Llzaj . With t from (23.19) one finds

(25.24)

with _ t/lT v qjo(O) v t/I/

~ OJ/(O) - (t/lT v t/lj t/ly, V t/I/)1/2' (25.25)

The usefulness of this expression lies in the fact that the quantities of interest are scalars rather than matrices. This facilitates the discussion of the qualitative behaviour of the dielectric susceptibility.

Example: Isotopic defects in alkali halides

F or alkali halides the summation over branches in the expression for ~ 0

is reduced to a single term for a TO mode polarized in the a = a' direction. For a mass defect without charge perturbations, one finds from (25.13),

(25.14), (25.17) that the susceptibility can be written as

V ()= Z6 ~( Zo )2 w2Llm (2526) N Xaa W J.I.[wio-(w+is)2] + N Mo[wio-(w+is)2] 1-w2 Llmgo' .

where Mo and go are the defect-site elements of the host-mass and Greenfunction matrices. For a resonance, (25.26) reduces to

(25.27)

with an integrated intensity of (cf. (25.21'))


where M 0 and Z 0 are the mass and charge of the replaced host ion. The total integrated absorption is from (25.7') proportional to

V OOs II 2 Z~ 11: Z~ t1m/Mo No X",,(w)dw =11:----;- N Mo 1+t1m/Mo'

c) Raman scattering. The theory of defect-induced first-order Raman scattering was developed by GUREVICH et al. (1964, 1965, 1966); SENNETT (1965); XINH et al. (1965); and XINH (1976b, 1968) following treatments of the Raman effect in perfect crystals. Generally, it is understood that the light is scattered from the fluctuations of the electronic polarizability ("electronic" Raman effect). The polarizability expansion coefficients (in terms of phonon displacements) are generally assumed to be independent of the frequency of the incident light, and as nonlinearities they have been interpreted in terms of anharmonicities (LouDON, 1964b, 1965). A corresponding treatment for defects has been given by MARADUDIN and PERETTI (1967). We will give a shell-model treatment somewhat along the lines of COWLEY (1964) in Sect. 27f.

Recently two modifications have created interest. First, with the incidentlight frequency close to an electronic transition, the polarizability no longer is a constant but may become very large. This is called the resonant Raman effect (R.M. MARTIN, 1971; BENDOW and BIRMAN, 1971; and references therein). In recent years the resonance Raman effect has been investigated experimentally and theoretically, see Sect. 25d.

The other modification is due to the ionic polarizability in addition to that of the electrons. The ionic polarizability may playa role for incident light with low frequencies at which the expansion terms again may no longer be constant. The ionic Raman effect in perturbed crystals has been discussed by MARADUDIN and WALLIS (1970) and HUMPHREYS et al. (1972). The Jahn-Teller system CaO: Cu2+ has been investigated experimentally and theoretically by GUHA and CHASE (1975).

The double differential scattering cross-section (per unit solid angle dOs and frequency interval dO) has been given in Sect. 9 as a function of energy transfer O=wi-ws as

d2 (J' w 3 w. d n dn=2s ; L e"(pJep(ps)f.!i<>x"pi<>X,,'p,)e,,,(pJep'(ps)· ~"s~" 11: C "p,,' p'

(25.28)

The most important quantity here is the spectral density function fo of the fluctuations <>X of the electronic susceptibility around a time-independent value. In Sect. 7 the susceptibility was written as <>X"p(Q) where Q was the lightmomentum transfer. The other quantities have the same meaning as in Sect. 7. The spectral density function fo is conventionally written as (Born and Huang, 1954)

(25.29)

One now expands the electronic susceptibilities in terms of phonon normal coordinates, (25.30)

The zero-order term contains only spatial inhomogeneities and is the ordinary high-frequency susceptibility. Inserted into (25.28), the zero-order terms give a


frequency-independent contribution to the cross-section and describes Rayleigh scattering. This will not however, be pursued any further here. The first term which is of interest here is the one in which both susceptibilities are expanded into the first order for which one finds in the harmonic approximation, cf. (34.75) and (34.50),

" 2ea). J g(JXo:pIJXo:'p,)=2L.l!p(A)1!'p,(A)Im 2 (Q . )2 (1+ng) ). ea).- +le

= 2h L l!p(L, y) 1!, p,(.C, y') Gil (L, y; L, )1')(1 + ng) LV)')"

=2h~~G"~'p,(1+ng), (25.31) where

l!p(A) = L l!p(L, y) X(L, Y I A)(h/2ea).)1/2 Ly

and l!p(L, y) are the first-order expansion terms of the susceptibilty in terms of normal coordinates and real-space displacements, respectively.

The higher-order expansion terms lead to higher-order Raman spectra in which one is not interested at the moment; second-order spectra will be mentioned in Sect. 27 f. The relation of the polarizability expansion coefficients to the photoelastic constants has been investigated by BENEDEK and coworkers (BENEDEK and NARDELLI, 1967a; BENEDEK and MULAZZI, 1969a, b; BENEDEK and TERZI, 1971, 1973) and LEIGH and SZIGETI (1969, 1970).

In (25.31) the first-order Raman spectrum has the same form as the firstorder infrared spectrum as given by (25.6). In general, there will be a contribution to the scattered intensity from the lattice and a defect-induced part. The lattice contribution is from the Raman oscillators which play the same role in Raman scattering as the Reststrahlen oscillators do in infrared absorption. One proceeds, therefore, exactly as in Sect. 25b to obtain the result for the Raman cross-section, (25.28), (25.29) with

io:pyo(Q)=~Im {p~p [~o(O) - ~ ~o(O) t~o(O)] Pyo

(25.32)

and

(25.33)

where the only surviving terms in ~o(O), (25.9), are those for which the branch indexj refers to Raman-active oscillators. In alkali halides, for which ~o(O) Po,o:p is identically zero, only the last term of (25.32) gives non-zero contributions,

iO:/lyo(Q)=~ ~ Ap~pgll Apyo(1 +ng). (25.34)

This equation is analogous to (25.22) for the infrared absorption due to defects in group-IV semiconductors. Clearly, the defect has to introduce a change in the


first-order polarizability tensor in order to give rise to defect-induced first-order Raman spectra. On the other hand, defects in crystals with Raman-active oscillators induce first-order spectra even without induced polarizability change. In this case the additional scattering intensity is given in terms of

iapyb(Q) =~ ~ p6.aP ~ 0(0) v gil V ~ 0(0) Po, yb(1 + nn) (25.35)

which is the analog of (25.20). The arguments for resonance modes in the spectrum are as in (25.21).

The main difference between the infrared and Raman spectra is that the latter have a temperature-dependent intensity. For T=O K the total integrated scattered intensity 6 is from (25.31) proportional to

S iapyb(Q)(1 + nn)-l dQ2 = h~~ M- 1 ~b o

as in (25.8), which reduces, for crystals without Raman oscillators, to

S iapyb (Q)(1 +nn)-l dQ2 =hLlp~p m- 1 LlpYb' o

(25.36)

(25.37)

Since the mass of a substitutional single defect (not defect pair) in alkali halides is not involved in the even-parity vibrations (which scatter the light), one finds that the total intensity of these spectra is usually completely independent of the dynamical perturbations and depends only on the polarizability changes. The shell-model treatment in Sect. 27 f will modify this result.

Example: Raman scattering by substitutional defects in alkali halides

Expressions like (25.34), valid for alkali halides, have been obtained by KLEINMAN (1964); XINH et al. (1965); SENNET (1965); and others, see, e.g., the work by HARLEY et al. (1971) and references therein.

Symmetry arguments show that in crystals with the rocksalt structure there are only three independent elements of the scattering tensor,

iaaaa=ill' iaapp=i12' iaPaP=i44 (a=t=f3).

With only nearest-neighbor interaction, there are three different non-zero terms of the first-order polarizability,

Pxx (100,x)=a, Pxx(OlO,y)=b, PXy(lOO,y)=c

and those resulting from the group operations of the group 0h' In terms of the symmetrized displacements of the irreducible representations of the group Oh' one has from Sect. 22 d

2h ; ill (Q)+2i 12(Q)=- (a+2b? g"(I;+)(1 +nn),

n

6 Actually, the intensity multiplied by Q.

344

with

and


g(r) = go(r)(l + vCr) gO(r))-l,

go (I;+) = go(100, x; 100, x) - go(100, x; 100, x) + 4go(100, x; 010, y),

go(I;~)= go (100, x; 100, x) - go(100, x; 100, x) - 2go(100, x; 010, y),

Sect. 25

(25.38)

go(I;~) = go(100, y; 100, y) - go(100, y; 100, y) + 2go(100, y; 010, x), (25.39)

(25.40)

as in Sect. 22d. Llfll and Llf~ denote the change in the longitudinal and transverse force constants between the defect and its nearest neighbors, which are assumed to be the only force~constant changes.

HARLEY et al. (1971) have found that their TI + -induced spectra can be described very well by unperturbed Green functions. This allows the conclusion

Fig. 25.3. Experimental and theoretical Raman spectra for KBr: TI +: above E component, below Tzg

component. The temperature of the experiment was 15°K and the instrumental gain was the same for both doped and pure crystals and for Eg and T2g spectra. The theoretical curves are for zero force-constant change and were computed using 0 K Breathing-Shell Model and 90 K Shell Model (VI) phonons. They have been normalized to reflect the experimental intensities. (HARLEY et a!.,

1971 )

N

:3


that the Tl + impurity does not induce any essential change in the force constants. This view was opposed by BENEDEK and TERZI (1971, 1973) who found a relative weakening of the effective force constants of typically 20 %, and relaxation-induced stiffening of the force constants between the nearest and next-nearest neighbors in the (100) direction of 20 to 70 %. On the other hand, Benedek and Terzi's effective force constant for, e.g., KCI is about 20 % smaller than the shell-shell overlap force constant in the shell model to which the changes are compared by HARLEY et al.

In Figs. 25.3 and 25.4 experimental and theoretical results are compared for TI + -induced first-order Raman spectra in KBr. Both figures show the experimental data of HARLEY et al.; Fig. 25.3 shows theoretical calculations using the shell (SM) and breathing-shell models (BSM). Figure 25.4 contains calculations using the deformation dipole model (DDM). Peak positions vary from model to model, relative peak heights are the same in shell and breathing-shell models. A detailed study of the influence on the change in nearest-neighbor force constant is given in Fig. 25.5; this seems to rule out any change in excess of 10 %. The calculated T2g(r;~) main peak and Eg(~i) secondary peak at about 90 cm- 1

o 50 100 150 cm-1

KBr :Tl+

U1 N

E '!ju

'Q

<al~ N a "tJ"tJ

~IQ.

500

N U1

'" N , Q

'" :3

0.5

0

KBr:T1+ RT

\ ~ Pi T2g

\ ., I! Expt. ( Harley et., \./.~ . ..-' "Ii ! (I. 15 K

\ /\ .'"\..-/ ........ -

Calculated

Exptl.gap

~ 01--"""'---...1....- ~

0.5

a

Calculated (Isotopic

/ defect)

b

0.5

o

Calculated (4 NFC)

Fig. 25.4 a, b. Calculated and experimental squared-frequency projected density of states for Tl + in KBr (left scale) versus frequency. Calculated Raman intensities (right scale) versus frequency. (BE.

NEDEK and TERZI, 1971)

346

o


positive defect

85M Eg

40 eo 120

FREQUENCY (em-I)

VERTICAL I UNIT

160

Sect. 25

Fig. 25.5. Perturbed theoretical Eg spectra for KBr using 0 K BSM phonons. The relative vertical scales are indicated by vertical unit bars. (HARLEY et aI., 1971)

have too Iowa frequency in all models, but least in the breathing-shell model. Also, both the shell model and the breathing-shell model show about the correct Eg peak-height ratio, while the deformation dipole model is out by about 20 %. See also the calculations by KARO and HARDY (1975, 1976; HARDY and KARO 1976) and PAGE (1976) for comparison of the results using the different models. A comparison of experimental and theoretical results for KCl:TI+ was given in Fig. 21.3. In a subsequent paper, ROBBINS and PAGE (1976) have shown that inclusion of polarizability derivatives at the second-nearest neighbors of the defect brings the theoretical calculations into perfect agreement with experiment.

HARLEY et al. (1971) point out that a comparison like the one above shows that the shell and breathing-shell models are very similar. They do have minor


differences in the dispersion curves which show up as different peak positions, but the corresponding eigenvectors seem very similar as is indicated by the approximately equal peak intensities, although these are an integral of the amplitudes over a certain frequency range. Ag+ impurities and later a series of defects in NaCI and KCI crystals have been investigated by KAISER and MOLLER (1969); MOLLER and KAISER (1972); KAISER et al. (1973); MONTGOMERY et al. (1972a); and HONER ZU SIEDERDISSEN (1976); a pair vibration has been found also (MOLLER et aI., 1970). Cu + impurities have been studied by MONTGOMERY and KIRBY (1971); GANGULY et al. (1972); and in a special context by IVANOV and PINKEVICH (1975). U-center local-mode overtones have been found by MONTGOMERY et al. (1972b) and WOLFRAM et al. (1972).

Theoretical spectra of doped CsCI (MARTIN, 1972) show little change even with drastic changes of the force constants around the defect; therefore, Raman spectra of CsCI should provide information mainly about the host crystal.

Often model calculations with force-constant changes restricted to nearest neighbors are successful (BUCHANAN et aI., 1974a, 1975; for example), but this is not always so (CHASE et aI., 1973). It is interesting to note that in the TI + -doped potassium halides, where no or only weak force-constant changes are needed, the anharmonic effects can also be explained with the unperturbed anharmonic force constants (TAYLOR et al., 1975; see also HAQUE, 1975). These spectra have also been obtained by YACOBY and YUST (1972; JUST and YACOBY, 1978) using differential techniques.

In recent years Raman scattering has become a fairly standard technique, and it is impossible to cite all the relevant literature.

d) Resonance Raman scattering. So far, in the off-resonance Raman effect, the frequency of the incoming and scattered light is small compared to the frequencies of the involved virtual electronic transitions. When both frequencies become comparable in the resonance Raman effect special effects are observed. Only in recent years has the resonance Raman effect been observed in perturbed crystals. In these cases the external light is in (or near) resonance with electronic transitions connected with the defect.

Experimentally, the F center in alkali halides is the most thoroughly investigated example (WORLOCK and PORTO, 1965; BUCHENAUER et at, 1969; PAN and LOTY, 1976; BmSSON et al., 1976a, b, c, 1978; SADOC et aI., 1976; GLYNN et al. 1977; LEFRANT et aI., 1978). The F-center absorption is due to transitions from the 1s ground state to the np excited states leading to essentially two broad bands, the F band (1s~2p) and the weaker K band (1s~np, n ~ 3), as shown in the inset of Fig. 25.6 below. Resonance enhancement in crystals with molecular defects has been observed by MARTIN (1975a, 1976); BILL and VAN DER OsTEN (1976); and MARTIN and ONARI (1977); in ZnS crystals with various defects by NAT ADZE et al. (1976); ZIGONE et al. (1978); and KROL et al. (1978); in KI:TI+ by WALKER and BENEDEK (1979); and in CdF2 :In3+ by O'HORO and WHITE (1973).

A typical result for the F center in an alkali halide obtained by PAN and LUTY (1976) is shown in Fig. 25.6: As the incident-light frequency is tuned through the F band, resonance enhancement of the whole spectrum (not clearly


--Alg

---- Eg .•••••..•..• T2g

Frequency shift [cm-l]

o o iO

10

Sect. 25

Fig.25.6. The A 1g , Eg , and T2g components of the resonance Raman spectra at 10 K due to F centers in KCl for different incident-light wavelengths. The inset shows the position of light frequencies together with the optical absorption spectrum which consists of essentially two bands, namely the contributions from the electronic transition to the first (F) and higher (K) excited

states. (Data by PAN and LOTY as cited by ROBBINS and PAGE, 1977)

seen in this figure because of normalized intensities) and of the two-phonon versus the one-phonon contribution is observed. As the frequency is tuned through the K band enhancement of the optical-phonon contribution versus the acoustical-phonon contribution occurs.

ROBBINS and PAGE (1977, 1978) gave a theoretical interpretation of the resonance Raman effect of the F center using perturbed phonons, point-ion (and also Hartree-Fock) electronic wavefunctions, and linear electron-phonon coupling (in the strong-coupling limit). Results are shown in Fig. 25.7. The dependence of the spectra upon the light frequency reflects the details of the electronic states: position of the electronic energies, form and extent of the wavefunctions and interference effects. Different forms of the wavefunctions (point-ion versus Hartree-Fock) require somewhat different force-constant changes for a best fit. Thus, in order to account for the dependence of the spectra as a function of light frequency and frequency shift one must know the local dynamics of the phonons as well as of the electrons. Similar experimental

Sect. 25 Applications of Green functions: Phonon spectra in perturbed crystals

A1g Theory --- A1g Expt. -------

~ ~ ~~~~--~--~~~--~~~--~--~~--~~~--~--~~ c CIJ -C

"0

~ CIJ .... '0 u

Vl

Eg Theory Eg Expt.

Frequency shift [cm-1]

349

Fig. 25.7. Experimental and theoretical Atg and Eg resonance Raman spectra due to F centers in KCl (cf. previous figure, ROBBINS and PAGE, 1977)

and theoretical investigations were made by BUISSON et al. (1978), MULAZZI and BUISSON (1978), and GHOMI and BUISSON (1979).

In their calculations, ROBBINS and PAGE extended a formalism due to HIZHNYAKOV and TEHVER (1967; HIZHNYAKOV et a1., 1969) which relates the electronic optical absorption (i.e. essentially the imaginary part of the dielectric function) to the electronic Raman scattering (which is proportional to essentially the square of the difference of the dielectric functions at incoming- and scattered-light frequencies - rather than the derivative as for the off-resonance case).

This formalism has been used by TONKS and PAGE (1979) to work out a test for the answer to the question to which extent the standard approximations (Born-Oppenheimer, Condon, linear electron-phonon coupling) are applicable to a particular system. Given an electronic absorption spectrum in the frequency regime of interest, the Raman intensity (for various phonon frequency shifts) as a function of light frequency (Raman "profile") can be pre-


" W 2 [1155 em-I] Ul -C ::J

.ci Ci '-'

Ul Q)

'0 ~

a.

c W 1 [1525 em-I] d

E d 0::

400 500

AL [nm]

Fig.25.8. Optical absorption spectrum (upper panel) and theoretical (curves) and experimental (dots) first-order Raman intensities (lower panels) for three phonon frequencies as a function of

light wavelength for jJ-carotene (after TONKS and PAGE, 1979)

dieted without explicit knowledge about the electron-phonon interaction. Agreement (disagreement) with experiment leads to the statement that the standard approximations can (cannot) be employed in any theoretical description. An example (for agreement) is shown in Fig. 25.8.

WALKER and BENEDEK (1979), who investigated the Tl+ defect in KI, found a similar distortion of the optical versus acoustical part (and within) of the one-phonon spectrum as a function of light frequency. They explained this dependence by the crystal-field versus the exchange part of the electron-phonon coupling (the exchange and spin-orbit interaction partially lifts the degeneracy of the first excited electronic state of T1 +).

Sect. 26 Dynamics of lattices with interstitial or molecular defects 351

Other treatments of the resonance Raman effect in defective systems are given by GANGULY and BIRMAN (1969); BARRIE and SHARPE (1972, 1978; BARRIE et aI., 1972; BARRIE and CHOW, 1978); MULAZZI (1976; MULAZZI and 'TERZI, 1979); and ABRAM (1977).

26. Dynamics of lattices with interstitial or molecular defects

a) Formulation of the problem. The introduction of a molecule into a lattice changes the properties of a lattice like a monatomic defect does. But the internal degrees of freedom of the molecule add new features to the perturbation and to the defect-induced effects. Conversely, the internal vibrations of a free molecule are changed due to the coupling of the molecule to the lattice. In addition, torsional motion (libration) of the molecule in the crystal matrix occurs, while hindered rotations obviously cannot be described in the present framework of lattice dynamics.

The effects of molecular defects on lattice dynamics in a Green-function formalism were first described by WAGNER (1963). He applied the formalism to a linear chain (WAGNER, 1963) and to a Debye lattice (WAGNER, 1964a). Some later contributions have been made by BLAESSER et aI. (1968) with particular reference to a bee lattice. These authors start from lattice, interstitial, and interaction Hamilton operators to calculate the interstitial Green function, (26.4) below. JONGST and KURI (1971) applied the formalism to substitutional OHions in alkali halides. They also treated the dynamics of molecular defects by a variational procedure. More complicated molecules have been investigated by KISELEV and LIAPZEV (1974; LIAPZEV and KISELEV, 1974). The N02 molecule and its influence on the Raman spectra has been investigated by ZAVT (1977) and REBANE et aI. (1977). In particular, the Fano resonance (FANO, 1961), i.e., an internal mode of the molecule resonating with lattice modes may occur. If the coupling is strong enough and the damping sufficiently weak, a splitting occurs as observed in KCI:CN- (WALTON et aI., 1974; NICKLOW, 1976; MOSTOLLER and WOOD, 1976; WOOD and MOSTOLLER, 1977). Finite concentrations of defects have been considered by MURTAZIN (1971) and VAMANU (1972).

The Green function of the lattice with additional atomic degrees of freedom will be formulated below. Use of the perfect-crystal Green function (and the perturbations due to the defects) can be made by means of matrix-partitioning techniques (LEHMAN and DEWAMES, 1963; KLEIN, 1963; LITZMAN, 1966; and references therein). The standard procedure, appropriate to interstitials, will be presented in Sect. 26b. The results also apply to molecular defects, but a more physical formulation for these latter will be given in Sect. 26c.

Defect-induced spectra due to interstitial and molecular defects have been sketched in Sects. 20g and 20h. For experimental data the reader is referred to these sections.

Common to the two types of defects to be considered here (interstitials and molecules) is the fact that the number of degrees of freedom of the lattice is enlarged. For reasons of simplicity the treatment will be restricted to the case in which the crystal is perturbed by one interstitial or by one diatomic molecule, respectively. The number of degrees of freedom is then enhanced by 3. The


Ll¢oO I \ Ll¢hh

~

[:>( ~1\'~ .11M2

.11M) .11M2

A"\I"~ -----

a b

Fig. 26.1 a, b. Perturbations induced by (a) molecular and; (b) interstitial defects. Illustration of (26.5)ff

following notation will be used (cf. also Fig. 26.1). For an interstitial: let h label the 3sN host-lattice atomic degrees of freedom; let 2 label the interstitial with its 3 degrees of freedom.

For a molecular defect:

let 0 label the host-lattice atoms without the defect (3(sN -1) degrees of freedom);

let 1 label one of the molecular atoms; let 2 label the other molecular atom.

One writes the equations of motion for the perturbed crystal as

for the case of a molecular defect, and for the interstitial as

LU=(I1>hh- wZMh I1>hZZ ) (Uh) =0. I1>Zh I1>zz -w M z U z

(26.2)

The key quantity for calculations of phonon properties always is the lattice Green function

G=L- 1 =(I1>-wz M)-l

which will be evaluated using the Green function of the perfect lattice.

b) Standard procedure - application to interstitials. In evaluating the Green function, L- \ use is made of the ideal-lattice Green function, L7/. The standard


procedure for evaluating L- 1 therefore is as follows. For notational simplicity let

with A and D a 3sN x 3sN and a 3 x 3 matrix, respectively. These matrices stand for submatrices occurring in (26.1) and (26.2). One finds

L-1 = ( 1 ) (A -BD-1 BT)-1 (1 -BD-1)+ (0 0). (26.3) _ D - 1 BT '0 . D - 1

Comparing this with the results for a lattice with a point defect,

L=A, L- 1=A-\

one can see that the (h, h) element looks very similar to the point-defect result, cr. (23.1). The (2,2) element in analogy to the (h, h) element can be written as

(L- 1b =D- 1 BT(A -BD- 1 BT)-1 BD,-1 +D- 1 =(D-BT A -1 B)-1 (26.4)

with a similar modification of the quantity D- 1• D- 1 can be looked at as the Green function of the Einstein oscillator represented by the particle (2). The frequency of an interstitial will be different from the Einstein-oscillator frequency since the interstitial is not vibrating in an absolutely static background. Instead, the surrounding of the defect is dragged along with the motion of the defect and hence the frequency becomes shifted. This effect is represented by the term BT A -1 B in the expression (26.4). If, on the other hand, the frequency of the intersitital is sufficiently high, the lattice can barely follow. The term BT A -1 B is small since A -1 =(tPhh-W2 M h)-1 is decreasing in its absolute value with increasing frequency.

However, this is not necessarily always so, and certain molecular defects, e.g. CN- in alkali halides, show a high-frequency stretching vibration in which both atoms of the molecule take part. The picture of an atom as an Einstein oscillator breaks down and so does the physical interpretation of (26.3). One will find a more physical representation for (26.3) for molecular defects later. To proceed from (26.3) on one separates pure-crystal quantities and perturbations,

(26.5)

and

(~O 0 0) (0 0 2M 2 0 Ml 0 2 0 LlMl

W =W +W

00 0 0 0

(26.6)

A few comments on the perturbing quantities, which are illustrated in Fig. 26.1: LlMl is the change in mass of the first molecular atom (particle (1» which is

zero for the interstitial case;


LI M 2 is the mass of the additional particle (2), i.e. of the second molecular atom or the interstitial;

Ll4»22 is the restoring force on the additional particle (2); Ll4»20 and Ll4»21 represent the coupling of particle (2) to the lattice (possibly

perturbed by particle (1)); Ll4»oo is the change in the force constants within the lattice due to

a) distQrtions around the defect, b) different restoring forces due to coupling to particle (2).

For an interstitial Ll4»lo, Ll4»Ol' and Ll4»l1 are of the kind of Ll4»oo; for a molecular defect Ll4»l1 and Ll4»10 is analogous to Ll4»22 and Ll4»20' respectively.

Using expressions (26.5) and (26.6) the terms appearing in (26.3) can be

written as ( 1 0) C~-l BT)= 0 1, (26.7)

. -(Ll4»22 _w2 LlM2)-1(Ll4»20' Ll4»21)

(A -BD- 1 BT)-l = [(4)>00-w2 Mo 4»012 )+ (Ll4»oo Ll4»o; ) 4»10 4»11 -w M1 Ll4»10 Ll4»l1 -w LlM1

(Ll4»02) 2 -1 ]-1 - Ll4»12 (Ll4»22- W LlM2) (Ll4»20,Ll4»21)

= [Lo + LI L+ LlE]-l

= [1+LQ1(LlL+LlE)]-1 L(/ = [l+Go V]- l GO' (26.8)

The simplification for the interstitial case obviously is

(-~-1 BT) = (-(~4»22 - w 2 LI M 2)-1 Ll4»2J

(A - BD- 1 BT)-l = [(4)>hh - w2 Mh) + Ll4»hh

- Ll4»h2 (Ll4» 22 - w2 LIM 2) -1 Ll4» 2h] -1.

(26.9)

(26.10)

The (01, 01)=(h, h) part of L- 1 looks like the Green function of a lattice with a point defect. However, the imperfection here brings in two changes, namely that due to particle (1), LlL; and the modification due to coupling of particle (2) to the lattice, LIE.

Now, in principle, one is able to calculate the Green function of the defective lattice, L- 1, from that of the pure crystal, Lo 1. In fact, for lattice dynamics of crystals with interstitials, (26.9) and (26.10) seem to be the appropriate formulation of the formalism.

The present formulation is unsatisfactory (even though correct) for molecular defects since one treats one of the molecular atoms quite differently from the other one. The selection of one molecular atom as being a defect (particle (1)) and the other as an additional degree of freedom (particle (2)) is quite arbitrary and unphysical as long as the molecule behaves like a molecule, i.e. shows internal vibrations reasonably well distinguished from the center-of-mass motion. A transformation of the equations of motion shall be performed which accounts for exactly this distinction between internal and center-of-mass motions.


c) Formalism modified for molecular defects. Considering the OH- molecule (for more complex molecular defects see the selection of references in Sect. 26a) substituting for a halide ion in alkali halides as an example, one finds that the internal vibrations (two degenerate librational modes, one stretching mode) have frequencies well separated from the phonon bands. The lattice is unable to follow these high-frequency vibrations, and therefore the lattice modes are essentially decoupled from the molecular internal modes. However, the lattice modes will be changed by the motion of the molecule as a whole since its mass and its coupling to the lattice in general will be different from those of the original halide ion.

One constructs a transformation which has the following property. When applied to the displacements of both molecular masses, it should lead to the displacement of the molecule's center of mass and to a relative displacement between its two masses. Application of this transformation, S, to the mass matrix M which appears in the equations of motion should result in the total mass, M I ,

and the reduced mass, M r , instead of Ml and M 2 ,

with

One finds that

M I =M1 +M2

M r =M1 M 2 (M1 +M2)-1.

serves this purpose. The equation of motion

now reads

with

One also has

just as expected.

LI=O

SLI=SLST(ST)-ll=O

Using the orthonormality conditions (22.6a),

LM~/21·(A)11(A)Ml/2=lbi j for i,j=O,1,2 A ~ ~ J J .,

(26.11)


one finds the same relation holding for i, j = 0, t, r, of (26.11), since

(SMST)(S X)(X t S~ 1) = S(M X xt)S~ 1 = SS~ 1 = 1.

For the matrix L one obtains after transformation

Sect. 26

M (L10_ L20) M (L11 +L12 L21 +L22) M2 (L11_ L12+L21 L22) r M1 M2 r M1 M2 r Mi M1M2 + M~

To interpret the result, one separates force constants and masses

Lij= iPij -W2 Mibij'

One further uses the infinitesimal translational invariance of the crystal to express the self-interaction terms iP 11 and iP 22'

iP 11 =- IiP 1j=-iP12 - I iP 1j , j*l j*1,2

shortly written as

and analogously for iP 22' One then obtains

with

(iPoo

SLST = iPtO

iPrO

The elements can be interpreted as follows:

(26.12)

describes the coupling of the molecular unit to the lattice;


(t, t) - 'L(<<P01 + «P02) then is the force constant for the restoring force of the molecular unit;

(r,r) -«P12 is the force constant for the coupling of atom (1) to atom (2). For a free ion this would be the only force constant; for an ion embedded in a crystal, coupling to the lattice gives additional terms, the form of which is not obvious, as is that of the (r, t) element.

Of course, the masses turn out as desired. One now has treated both molecular atoms in a symmetrical way and has

thereby separated the motion of the molecule's center of mass from its internal vibrations. It is now an easy task to find the Green function

G=L- 1 =(S-l SLSTS)-l =ST(SLST)-l S

where one simply uses (26.3) and (26.5) to (26.8) with the indices 1 and 2 replaced by rand t, respectively, for evaluating (SLST)-l.

It should be mentioned that (26.4) now reads

Lzi = [D _BTA -1 B]-l = [(<<Prr-ui Mr)-«Prh(<<Phh _w2 Mh)-l «Phr]-l.

The pole of D- 1=(<<Prr -w2 M r)-1 gives the frequency wr=VcPrr/Mr of the free molecule. As 1 A -11 is negative for frequencies above the phonon bands, the pole of [D-BTA -1 B]-l =[Mr (W;-w2)-1_BTA -1 B]-t, giving the corresponding frequency of the molecule in the crystal, is shifted to higher frequencies due to coupling to the lattice, if one assumes «P rr to be the same as that of the free ion. However, a glance at (26.12) shows that «Prr in fact is different from the free-ion force constant. However, for the stretching vibration one can assume «POl and «P02 to be positive (as e.g. from a Born-Mayer potential) which leads to an even more enhanced frequency.

To recall the assumptions made about the molecular defect, the usual latticedynamical assumptions (existence, expandability of potential, etc.) imply that the molecule is oriented along a certain direction for an appreciable time, so that a force constant is a meaningful thing to talk about. Otherwise the defect is rotating and the concept of small lib rations breaks down. (The spacing of levels is totally different in the two cases.)

The modified version of the formalism aims at molecules which represent a "unit"; this, however, does not put a constraint on the defect since it is perfectly general. This, for example, means that the stretching-vibration frequency need not necessarily be above the phonon bands.

This formalism is written up for diatomic molecules; otherwise nothing has been assumed about the symmetry of the defect in the crystal. That enters the calculation when the force constants «POl' «P02 are parameterized.

For practical calculations one will have to assume that the interaction of the defect with the crystal will not be of too long range; to be a little more precise, one assumes that the coupling of the defect to the lattice, Ll «P 1 0 + Ll «P 2 0' will not be appreciably different from the coupling of the original atom, «P 1 0, for longer distances as reasoned in Sect. 22e. For the same reason the charges on the molecular atoms will have to be close together, so they will not form a large


permanent dipole moment, and they will have to be essentially at the site of the center of charge of the substituted ion. If long-range changes appear, the Lifshitz formalism becomes impractical. The Lifshitz formalism may therefore not always be applicable to real systems.

27. Shell-model treatment of the dynamics of perturbed lattices and the model theory of infrared-absorption and Raman-scattering spectra

a) The use of shell models. The use of shell models of various kinds for the description of phonon dispersion curves has been widely accepted. As discussed in more detail in Sect. 4, the internal (deformation) degrees of freedom simulate electronic excitations which take place while the atoms or ions move. The shell models, therefore, present a classical description of a limited number of quantum mechanical phenomena. It is hoped that the phenomenological model properties support the recognition as to what the most important quantum mechanical electronic excitations are, and that this in turn will give insight into the qualitative and quantitative aspects of the forces which bind ions or atoms in a lattice structure. Ideally, one would have to calculate the perturbed dynamics and coupling constants from quantum-mechanical principles, as has been initiated already (see the references in Sect. 33).

Common to the shell models is the adiabatic approximation which allows all electronic degrees of freedom to be eliminated. This enables "model" quantities (i.e. force constants) to be grouped together (see (4.23), for example) so that the quantities involved can be interpreted as "formal" or "effective" coupling constants (Sect. 27b) and "effective" charges. These are no longer constant quantities but depend on the wave vector of the phonon and thus implicitly on the phonon energy. (We will take up this aspect in Sects. 27b and d.) In pure crystals this has the desired consequence that the phonon energies are affected to a different extent for wave vectors in different parts of the Brillouin zone. For example, the breathing degree of freedom is most influential around q =(1,1,1) nlro and least around q=(O,O,O) and q=(1,O,O)nlro for the NaClstructure crystals.

Introducing a defect into the crystal will in general lead to a polarizability change at the defect site due to a change in an "electronic degree of freedom". One can easily imagine that the elimination of this perturbed degree of freedom would result in an exceedingly complicated expression for the "formal" perturbation matrix V (Sect. 22c). One will see below (Sect. 27c) that the threatening complexity is circumvented by applying the Lifshitz formalism to the model equations of motion (i.e. those in which the electronic degrees of freedom explicitly occur) rather than to the "formal" ones (in which the electronic degrees of freedom are eliminated). This was first proposed by PAGE and DICK (1967); see also BILZ et al. (1966).

One may argue that in some cases the electronic degrees of freedom playa minor role for the lattice dynamics, anyway, and that a small perturbation of the electrons results in negligible effects. This is opposed by considering the model expression for the Reststrahlen frequency of a diatomic crystal, see

Sect. 27 Shell-model treatment of the dynamics of perturbed lattices 359

(4.21), 2 I 4n 2/ ( 4n) /1WTO =Ro - T Z S I-T IX,

(COCHRAN 1960) where IX is the electronic polarizability, R~ is an effective short-range force constant and Zs is an effective charge (all in appropriate units). The formula shows the separate contributions from short-range and Coulomb forces. The denominator

I-IX 4n/3 = 3/(200 + 2)

is typically of the order of 1 for the alkali halides. In the (nearly) ferroelectric materials SnTe (COWLEY et aI., 1969) and PbTe (COCHRAN et aI., 1966) the denominator is of the order of 0.1, the contribution from the Coulomb forces becomes so large that it (nearly) cancels the contribution from the short-range forces leading to a (nearly) soft mode. This means that all quantities in the expression for wTO are large, and that near-cancellation occurs. By introducing defects into the crystal one expects strong changes in the dynamics of the Reststrahl en mode even if the perturbations are very small. Defects may change R~, Zs, IX, or all; but the most crucial quantity seems to be the electronic polarizability IX, whence a somewhat more detailed treatment (than given in the following) of the electronic degrees and the influence of defects on their properties seems necessary. On the experimental side, a strong dependence of the ferroelectric behaviour upon defect concentration is observed, see e.g. YACOBY and YUST (1973), YACOBY et al. (1977), TAYLOR and MURRAY (1979), WAGNER et al. (1980), BAUERLE et al. (1980) for the perovskites, BURNS and SCOTT (1972), HISANO and RYAN (1972), GEBHARDT and MULLER-LIERHEIM (1978) for PbsGe3_xSixOll' to mention only a few, and the review articles by KAWAMURA (1978) and JANTSCH (1982) and the references therein.

From lattice dynamics it is known that the "effective" ("formal") charge is a frequency-dependent quantity even though generally measured at the Reststrahlen frequency only. Since the one-phonon absorption is proportional to the square of the linear dipole moment, i.e., to the ionic charges, any theory of defect-induced absorption should account for this frequency dependence. Here again, the "model" formulation of the dielectric susceptibility will be shown to account for the frequency dependence of the effective charge in a very easy way (Sect. 27 d).

Analogously, the "model" formulation of the susceptibility due to non-linear dipole moments (Sect. 27 e) as well as of the Raman scattering tensor (Sect. 27 f) will prove superior to the "formal" picture. These two features have two important aspects. One is connected with the model interpretation of these quantities by means of virtual electronic transitions. This has been the subject of Sects. 2b and 4d, and it will be taken up again in a moment. The other concerns the symmetry of electronic excitations. Suppose one adds a defect with a large probability for transitions with tetragonal symmetry (like Ag+) to a lattice; then, as is discussed in detail in Sects. 4d, 18, and 27f, strong defectinduced first-order Raman spectra with tetragonal symmetry are to be expected.


Here, the usefulness of the study of defects on the one hand and of Raman and infrared spectra on the other becomes apparent: Certain features in these spectra (like strong tetragonal symmetry components) are indications for the symmetry of virtual electronic excitations of the defect at least, or, in a wider sense, of the whole (defective) lattice. This information, in turn, can be used for developing lattice dynamical models (for example, for an AgCllattice where the Ag+ ion under consideration forms a sublattice).

Not only may new models be developed with the knowledge furnished by the interpretation of the spectra, but also existing models with deformable ions may be improved. For example, the electronic susceptibility, which is directly related to the electronic polarizability, is in shell-model terms given by

(27.1)

(see Sect. 27 d below), where Zs contains the shell charges and where lPss is the shell-shell force-constant matrix. In the simplest form of a model theory there is one parameter for each of the two quantities. The expression above may be calculated from essentially one experimentally known quantity, the electronic polarizability. This leaves one with some ambiguity - within certain limits - in the choice of parameters for the shell charge and interaction strength. One will see below that the first-order Raman tensor, for example, contains - among other things - these two quantities in a different combination. Raman spectra may, therefore, give information not only on the type of electronic excitations but also on some quantitative aspects of these excitations.

The other above-mentioned aspect of the Raman scattering tensor and of the dielectric susceptibility due to higher-order dipole moments concerns the (nonlinear) coupling between light waves and phonons and the parameterization of this coupling. The light couples to the lattice via the charges within the lattice. These may be the ones moving with the ions or those in the high-frequency motion of the electronic "shells" (in shell-model terms). Excitation of this highfrequency motion by the light represents a real or virtual electronic excitation. The shell model takes account of the electronic excitations in a classical way by treating their dynamics like that of oscillators. Clearly, real transitions between electronic states cannot be described in this classical scheme. On the other hand, the classical picture may be sufficient to describe polarization effects, i.e. virtual transitions, for energies far enough from transition energies. It seems, however, that the shell model is restricted to ions with well-localized electrons. Contrary to this, donors and acceptors in insulators at low temperatures have bound electrons with wave functions extending far out into the crystal. For these cases a completely different dielectric Clausius-Mosotti-type approach was presented by BARKER (1973).

When an electron is virtually excited it may return to its ground state. This describes a polarization effect leading to an effective force constant and an effective charge. Alternatively, the electronic excitation may "decay" into two phonons or a phonon and a light wave, which provides a microscopic explanation of the absorption due to the second-order dipole moment and of the first-


order Raman scattering, respectively. In a shell-model treatment of these effects, one first assumes the electrons to be (virtually) excited like classical oscillators, and secondly one tries to allow the harmonic and anharmonic coupling constants between cores and shells to be independent of whether the excitation of the shells is by a phonon, in which case the shells follow the cores adiabatically, or is a (virtual) electronic excitation. It is hoped that a model interpretation along these lines will supply a unified description of phonon lifetimes, higherorder dipole moments, Raman scattering intensity, and possibly other features.

For virtual electronic transitions one can find a propagator approximately described by classically moving electrons. A graphical representation of various quantities to appear below has been given in Tables 2.1 and 2.3.

Once the electron-phonon interaction is introduced for a microscopic interpretation of the Raman scattering tensor, the vibronic sideband spectra of electronic transitions contain further information about lattice dynamics. Vibronic spectra are different from the Raman effect in that they accompany electronic transitions which are real and for which the shell-model interpretation would probably be overstressed. However, the electric field due to the ionic vibrations as it will appear below is indeed an important quantity in the theory of vibronics.

Experimental vibronic sidebands accompanying electronic transitions of Sm + + impurities in alkali halides and the theory of vibronics have been presented in a series of papers by BRON, WAGNER and coworkers (BRON and HELLER, 1964; BRON, 1965, 1969; BRON and WAGNER, 1965, 1966, 1968; WAGNER, 1964b, 1965, 1967b, 1968b; WAGNER and BRON, 1965; KOHNER and WAGNER, 1970, 1972; KOHNER et aI., 1972; BRON, 1975), previously measured by KAPL YANSKII and FEOFILOV (1962, 1964) with later contributions by the McMASTER research group (TIMUSK and BUCHANAN, 1967; TIMUSK et aI., 1968; BUCHANAN and WOLL, 1969) and, e.g., IGNATEV and OVSYANSKIN (1977). A spectrum of CaF2 :Eu3 + has been observed by Ward and Whippey (1974). Other systems have been investigated by IGNATEV and OVSYANSKIN (1976), NORAS and ALLEN (1979), and others. Phonon spectra are also seen in the wings of zero-phonon lines due to transitions occuring at F -center complexes (BEAUMONT et aI., 1972; see FITCHEN, 1968, for a review).

b) Effective force constants. A common procedure for treating the dynamics of defects is the following: One usually represents the unperturbed crystal by some kind of model with deformable ions. For the defect, on the other hand, one takes an undeformable ion which interacts with the lattice by coupling only to the core of the lattice ions via some "effective" force constants. This, in fact, may be not too bad an approximation if the defect and the lattice ions are weakly polarizable, even though it lacks a unified concept. In general, the question is: what are the forces between the defect and the ions, i.e. what are the "effective" force constants?

A simple example is the linear symmetrical three-atomic molecule as shown in Fig. 27.1. It has two different eigenmodes in the direction of the axis. The frequency of the even- and the odd-parity vibration is given by


Fig. 27.1. Linear symmetrical molecule with one polarizable ion and a "long-range" force

Compared with a simple model of just three masses connected by two springs with spring constants ii' the two vibrations in the more elaborate model have effective force constants

which are different for the two different kinds of vibrations. Long-range forces (f2) and polarizabilities (g) may be important for different vibrations and in different ways. The same is true of a crystal with a defect. The even-parity modes are independent of the polarizability of the defect, while the odd-parity modes depend on the short-range and electrical polarizability. A breathing deformation of the defect will result in an effective Tl force constant which is different from the T12 one. So in principle, if one does it at all, one would have to define an effective force constant for every single symmetry. Not surprisingly, in many cases it is found that the force-constant change for a specific system has to have different values in order to reproduce spectra of different symmetry (MONTGOMERY et a1., 1972a; for example); it is not a priori clear, however, whether that is an effect of ionic deformabilities or just a property of the particular defect model.

BENEDEK and NARDELLI (1967b, 1968b) suggest that effective force constants be defined from a resonance at frequency w=O (instability) for various symmetries. If one does so, one finds different effective force constants for different symmetries. Also, the relative changes in effective force constants for low-lying resonance frequencies have the desired value of close to 100 %. In a shell model, for comparison, the instability occurs for a change in the nearestneighbor shell-shell central force constant of about 60 % (PAGE, private communication). The effective-force-constant concept as defined by the instability condition seems, valuable, if at all, only for vibrations with very low frequencies.


It is known from lattice dynamics (Sect. 4) that in the adiabatic approximation effective long-range forces are created by eliminating the electronic degrees of freedom. The effective force constant driving a defect is, therefore, a sensible quantity only for an Einstein oscillator resonance. In general, polarization effects have to be taken into account separately for every single mode. In fact, BENEDEK and NARDELLI find different effective force constants for low- and high-frequency oscillations (see also KLEIN, 1968a). Even though effective force constants may be useful in estimating properties of very low- or high-frequency resonances, these force constants cannot themselves be estimated but need quantitative calculations using Green functions. The shell-model treatment of lattice and defect to be presented below not only avoids defining different effective force constants for various frequencies and symmetries but also supplies a unified model for both.

c) Shell-model extension of the Lifshitz formalism. The practical success of the Lifshitz formalism is closely connected with the fact that the perturbations introduced into the crystal are limited to a small region, the so-called impurity space. With increasing size of the impurity space, the Lifshitz formalism becomes more and more impractical. Two essential conditions underlie the assumption that the defect space is small (Sect. 22e): 1) The distortion does not extend appreciably far into the crystal. 2) There are no changes in long-range forces.

One will have to keep the first assumption here. As far as the second is concerned, there are two kinds of long-range forces. First, there is Coulomb interaction between the ions of an ionic crystal. As long as the impurities have the same charge as the ions they replace, the Coulomb interaction will not be changed. However, as discussed in Sect. 4, the rigid-ion model or the Kellermann model is a relatively poor model of the lattice dynamics of pure alkalihalide crystals. Important improvements are obtained by including internal degrees of freedom of the atoms, most importantly their polarizability, which is accounted for by the shell model and its modifications.

As also discussed in Sect. 4, the shell model has, besides the Coulomb forces, effective long-range forces: When an ion is displaced, the shells adjust instantaneously throughout the crystal and pull at all other atoms. If now the defect is displaced, the effective force exerted by the electrons is changed since there is a changed force between the defect and, say, nearest-neighbor shells.

In addition, the defect itself may be polarizable, the polarizability involving the shell charge. As the polarizability of the defect will be different from that of the replaced host ion, this usually means that the shell charge will be different even though the total ionic charge may be unchanged. This then necessarily leads to perturbed long-range Coulomb interactions.

There are, of course, defects whose charge is different from that of the host ion, requiring charge compensation for reasons of charge neutrality. For reasons to be discussed below, one has to exclude these kinds of impurities (the Sm + +

ion in alkali halides is a typical example) unless the compensating charge is close to the defect so that the long-range Coulomb field is that of a dipole rather than of a monopole and thus falls off much faster.


Finally, the electronic structure of the defect may be so different from that of the replaced host ion that electronic excitations, say of quadrupole-type, may be important and therefore have to be taken into account in addition to certain host lattice excitations, say, of dipole-type, as in alkali halides. Apart from the fact that our knowledge of the kinds of important excitations is rather poor, this will complicate matters. However, a very simple example of an additional internal degree of freedom will be given which, in essence, can be handled with methods similar to those described in Sect. 26.

The most successful and common model for lattice dynamics is the shell model. The results derived below can easily be generalized for the breathingshell model which has proven extremely successful for alkali halides.

As usual, one starts out from the shell-model equations of motion (cf. Sect. 3)

( iPee- w2Me iPes2 ) (Ue ) =0, iPse iPss - w Ms Us

iPPQ=iP~Q+ZpCZQ P, Q=C, S. (27.2)

It is irrelevant for the following whether Us denotes the absolute or the relative shell displacement or something in between (RA, 1972).

The matrices for the interactions between cores (index C) and shells (index S), iPPQ ' are made up of short-range (say, Born-Mayer) forces, iP~Q' and long-range Coulomb forces between the charges Zp of the particles. C is the Coulomb matrix. In the adiabatic approximation the mass of the shells is set equal to zero; the displacements of the shells can then be eliminated, and one obtains

(27.3) with

(27.3 a)

iPF is a "formal" (effective) force-constant matrix. One must distinguish between the formal force constants and the force constants contained in iP~Q' which will be called "model" force constants since they can be visualized as spring constants in the model.

It is now clear that a change in the long-range part of the potential will lead to a non-localized perturbation matrix. Furthermore, a change even in the shortrange repulsion part of the "model" shell-shell interaction matrix iP~s will lead to a long-range change in the "formal" force-constant matrix iPF through the inversion procedure. (About the inversion of the Coulomb matrix C see RA, 1972.) To avoid these difficulties one rewrites (27.2) as

The last of the equations above can be written as

"F = C(Ze ue + Zs us),

(27.4)


and since U F multiplied by a charge gives a force, UF itself is (the negative of) the harmonic part of the electric field produced by the vibrating lattice particles. Using the extended space as in (27.4), changes in the model force constants or in the charges will result in a perturbation matrix

(,1 4>~c - co2 ,1 M C ,1 4>~s

V = ,14>~c ,14>~s

,1Zc ,1Zs

(27.5)

that is localized about the defect. In general, one has

-,1C- 1 =C- 1 ,1C(l +C- 1 ,1C)-l C- 1

which is of long range even if ,1 C is localized. In order for ,1 C to be zero one must require that

1) the lattice distortion which contributes to ,1 C is negligible for the reason given in Sect. 22;

2) the change ,1 Z = ,1 Zc + ,1 Zs of the total charge at the defect is equal to zero. If it were not, one would find non-zero diagonal elements in the perturbation matrix for reasons of translational invariance

,1 [Z(L) C(L, IX; L, /1) Z(L)] = - ,1Z(O) C(O, IX; L, /1) Z(L)

,1Z(O) = -Z(L) Z(L) C(O, IX; L, /1)Z(L)

=Z(L),1 C(L, IX; L, /1)Z(L)

while the off-diagonal terms are unchanged (STRAUCH, 1968). One now assumes that the two conditions set out above are fulfilled by the

defect. In the case where the defect has some kind of additional internal degree of

freedom (index D) which couples to core or shell motion through short-range forces ,14>cD' ,14>sD' or electric fields (produced by all vibrating lattice particles), or any combination of these, one would have

instead of (27.5) to be inserted in (27.4). Elimination of UD in the adiabatic approximation would give the same

perturbation matrix as in (27.5) with

(in obvious notation) subtracted from each of the respective nine terms in (27.5).


The unperturbed Green function in the extended space is

One readily obtains

with 7

G1PQ (W) = lop(wl - wZl)-l lbQ

Gocc=lodwl-wZl)-l lbc=(CPj; _WZ Mod-l,

lOF = C(ZOC loc + Z os los),

los = - CPOS~ CPOSC loc,

(0 0 0 ) (0 Gz = 0 (CP~ss Zos )-1 = 0

o Zos -C- 1 0

o CPos~

CZos CPos~

The perturbed Green function turns out to be

G=[I+Go VJ- 1GO

Z )_1 OC ZOS .

_C- 1

=[1+(G1 +Gz) V]-1(G 1 +GZ)=GL +GE

with the frequency-independent ("electronic") part given by

GE =[I+Gz V]- l GZ

(0 0

= 0 (CP~ ss + ,d cP~s o Zos+,dZs

as expected.

Sect. 27

(27.6)

(27.7)

(27.7 a)

(27.7b)

(27.7 c)

(27.8)

(27.9)

(27.9')

Dynamical properties of perturbed lattices, like local-mode frequencies or perturbed Green functions, can now be found from the equations above. Compared with the formal treatment this is at the expense of an enlarged, but still limited impurity space. On the other hand, if the shell model is to have any physical significance, then a larger number of degrees of freedom and of parameters (now describing polarizabilities as well) cannot in any case be avoided.

Polarization effects were first taken into account by FIESCH! et al. (1964, 1965), who allowed for a change in the shell charge as well as in the core-shell

7 <l)os1 is to be understood as (<I)oSS)-I.


interaction. Since FlESCHI et al. worked in a formal picture, approximations were necessary to reduce the size of the defect matrix. In fact, they took into account only the element of the defect site, which means that they assumed all cores to be fixed except that of the defect. This is a reasonable assumption for the Ucenter localized mode for which numerical calculations are done. The shells are free to adjust and are thus implicitly taken into account.

The extension of the Lifshitz formalism to explicitly include the motion of the shells was first suggested by PAGE and DICK (1967). Further extension to electric fields has been done by PAGE and STRAUCH (1967, 1968); PAGE (1969). The analogous treatment for defects in the bond-charge model was carried through by NIELSEN (1982). It is interesting to note that the (frequency-dependent part of the perturbed) Green function GLFF has direct applications in the theory of vibronics (see references in Sect. 27a; a shell model treatment can be found in papers by BRON, 1969; and BUCHANAN and WOLL, 1969; see also PAGE, 1969), where the electric field ( -IF) from the vibrating lattice modulates electronic transitions.

The Green function, Go, as given by (27.6) is composed of two parts, one of them, G2 , being independent of frequency. This property is to some extent due to the application of the Born-Oppenheimer approximation. For clarification of the meaning of this term, the Green function is rewritten with non-zero electronic masses,

with

4)~cs 4)~ss-w2 Mos

Zos

Z )_1 OC

Zos =Go, _C- 1

GoPQ = I lop(A)(wi-w2)-116Q(A)+G2PQ A~ AphAel

G2FF = -c,

(27.6')

all other submatrices of G2 being zero. One now has a set of quantum numbers describing lattice vibrations (Aph) and another set describing high-frequency oscillations of the electronic shells (Ae1) in an essentially rigid background of the cores (see below). The actual dispersion of the electronic motion does not reflect the electronic bands at all, since the electron motion is thought of as merely providing a mechanism of polarizability for the model. As a matter of fact, the shell model is constructed such as to fulfill the Clausius-Mosotti formula, which involves polarizabilities (see Sect. 27 d below).

It can be shown that I 10s(A)(wi - W 2)-1 I6s(A) (27.10) Ael

as in (27.6') reduces to (4)OSS)-1 as in (27.8) for vanishing electron mass (BRON and WAGNER, 1968): As the electron mass decreases, the frequencies wi increase as iPoss/Mos. The infrared frequency w is much smaller than W A and can be neglected. Since from the normalization (like (22.6)) xos~ Mol/2, (27.10) reduces to 4>oss, the mass dependence drops out. Since Xoc~MoJ/2,

L 10dA)(wi - W 2)-1 I6s(A) Ael


a _=~_t1_. __ .. ]-1

= [1 - -+- ___ .... ]-1_

= ----..--...-+ -+--------...--*--+ ....

Fig. 27.2 a, b. Graphical representation of the phonon Green function in a shell-model representation. (a): (27.11); (b): (27.6) for P, Q= C, S. Thick line, Goe6 thin line GbOt6 dashed line G2SS ; dots,

<Pocs and <Pose

tends to zero. These arguments convert the results as in (27.6') back into those as in (27.6) to (27.8). What one has learned hereby is, that (ePOSS)-1 is the remnant of an electronic propagator. Only the real part is preserved, since far-infrared frequencies are too small to induce any real transitions between electronic states (excitation of electrons, electron-hole pairs, excitons, plasmons, etc.). It should be noted that the phonon Green functions, G1PQ' implicitly contain the electronic propagator. Equations (27.7a) with (27.3 a), and (27.7) are interpreted in a graphical way in Fig. 27.2, where the appearance of the electron propagator is explicitly shown. The core-core Green function can be written (suppressing the index 0) as 7

Gee=(ePF _0)2 Md- 1

= (ePee - ePes ePs~/ ePse - 0)2 M e]-1

= [(G~t)-1 - ePes ePss 1 ePse]-1

= [1- G~t ePes ePss 1 ePse] -1 G~t. (27.11)

Here one has defined a rigid phonon propagator G~t, which is represented as a thin line in Fig. 27.2a. The dashed line represents the electron propagator and the dots the "coupling constants", ePes, between the electronic and lattice excitations. Goee, Goes, and Goss are shown in Fig. 27.2b in matrix form as in (27.7) (neglecting the electric field components). It appears that the excitation of a phonon via a shell displacement always occurs via an electronic excitation.

This formalism has been applied to the dynamics of the U center in alkali halides and the induced infrared spectra, see below, and to infrared and Raman spectra due to various defects in CsBr (HAQUE and STRAUCH, 1977; HAQUE et aI., 1977, 1978).

Example: The U center in alkali halides Quantitative calculations using the extended Lifshitz formalism were first

performed in connection with the U-center local mode (PAGE and STRAUCH, 1967, 1968) and its sideband spectrum (STRAUCH and PAGE, 1968; MACPHERSON and TIMUSK, 1970a). The latter will be discussed in more detail in Sect. 30j. The "formal" Lifshitz Green-function treatment of the· U-center local mode shows that the determinant of the matrix 1 + go v, (22.21), is equal within a few percent to the one of the defect-site elements of these matrices alone. Since considering only the defect-site elements is a very good approximation, the discussion of the shell-model treatment of the local mode is facilitated.


Fig. 27.3. Perturbations and their parameters in the shell-model representation of a defect

Treating the polarizable defect in a molecular model as in Fig. 27.3, with the nearest neighbors fixed, one obtains the equation of motion for a core displacement u and a shell displacement s,

wlmu=gu-gs

0= -gu+(2f+g)s

with the eigenfrequency given by

g is the isotropic core-shell spring constant and 2f = 2(f1 II + 2f1 J is the sum of central and non-central force constants of the springs which keep the defect shell in equilibrium. In an alkali halide g is typically one to two orders of magnitude larger than f To first approximation, one sets g ~ CX) and obtains for an Himpurity a value of 2f = 1.48 dynes/em. This value is listed as 2f4 in Table 27.l. Taking account of the polarizability results in a value 2f3 which differs by only 2.5 % if g is taken to be the unperturbed value, go' Keeping wlm fixed, the value of g may be decreased while 2f simultaneously increases. Since g is much larger than f, g can be changed over a rather large region without having any appreciable influence on the value off

This is, in fact, what one finds with the Green-function technique also. Figure 27.4 shows the results of the exact calculation. The dotted lines give the results if just the defect-site elements of g and v are taken account of. They correspond to


Table 27.1. Values of host and defect force constants (f.c.) in KCl:H-

f.c. calculated from f.c. value (104 dyn/cm)

2/01 =tWJio 2/02=2/011 2/03=2/011 +4/01- =Ro 2/04=R~

3.12 4.89 4.20 3.88

1.48

1.52

2.20

2.36

,dg/g-

0.5 1.0

a;=3.71 k=6.43 a;=14.4 I

==~ ..................... -... ~ ................... ~ .. ::::~ -0.4 xOO=36.4

8 I- KCI:H"" I ,dy-O r=34.S

--With breathing tX=32.4 XI .. 0.2

t 4 '- ---Without breathing ·········(0,0) Approximation tX=30.6 x

1 I L~=29.9 E u ...... c: >

"tJ I I

t «=0.372 «=0.546 «=1.02 ~ -~ ~ 04 ~ ............................................. ::::: .... ~=1.90 -

~ 81- KCI:H- ·······l<0C>=2.82

,dY=2.1S (Y=-l) J «=3.38 X\ .. 0.2

«=3.51 1 ~3.481

41- --With breathing ---Without breathing ......... (0,0) Approximation

I

o 3 6 Llg (105 dyn/ctn) __

Sect. 27

Fig. 27.4. Plots of A/l versus Ag for AZs=O and for Zs= -1 for V-centers in KCI. Included are the approximations of neglecting the nearest neighbor's breathing and of regarding only defect-site elements in the perturbation matrix. Values of IX, the polarizability of the defect, are given at various points along the curves. The A/>O and Ag>O portions of the curves have not been shown, as

discussed in the text. (PAGE and STRAUCH, 1967)


the solutions of the equations above, except for the fact that in the exact calculation all shells in the lattice are free to move, while in the above example all but the defect shell are fixed. Taking account of the motion of the neighbors does influence the result a bit, as is shown by the dashed curves in Fig. 27.4, whereas the inclusion of the breathing degrees of freedom has little further influence.

Since the defect vibrates in an essentially static lattice, hardly any electric field is produced at the defect site. Changing the charge on the defect core and shell has, therefore, relatively little influence on the local-mode frequency. The lower and upper parts of Fig. 27.4 correspond to two calculations in which the charges were either unchanged or altered to a value of the shell charge equal to one elementary charge. With varying values of charge and force constants, the polarizability (actually, the susceptibility) of the defect changes; the values are indicated at various points along the exact curves.

The curves shown in Fig. 27:4 are each one of two branches of solutions. The second branch extends into regio"ns with negative values of either f or g, and with mostly negative values of the polarizability. As it describes apparently unphysical solutions, the second branch is not shown.

Studying the local-mode frequency gives only limited information about the changes in the force constants introduced by the U center. There are other sources of information. For the specific example under consideration these are mostly sidebands of the local-mode absorption line. Their origin will be considered in more detail in Sect. 30j. For the present purpose, the results are as follows. The sideband shapes can be accounted for only if changes in the force constants between the defect and its first (Afl) and the first and the fourth neighbor (Af2) are taken into account. These changes are shown - among others - in Fig. 27.3. The change in the core-shell force constant Ag was shown to have little influence.

Using the force-constant changes Afl and Af2' as obtained from the Ucenter local-mode frequency and sideband, MACPHERSON and TIMUSK (1970b) calculated the far-infrared one-phonon defect-induced absorption spectrum. Their theoretical and experimental results are compared in Fig. 27.5a. The resonance-mode peak near 90 cm - 1 agrees quite well with the experimental value. MACPHERSON and TIMUSK have also studied the influence of various force-constant changes on the shape of the absorption curve. The results are shown in parts b to d of Fig. 27.5. As expected, the resonance peak shifts to lower frequencies if any of the force constants is weakened. The region around 83 cm -1 is a typical antiresonance region with a relatively high density of states. The resonance cannot move into this region with decreasing force constants. Instead, the resonance line intensity decreases while at the same time an incipient resonance at lower frequencies gains in intensity.

Otherwise the spectra look very similar. The assignment of numerical values to the various force-constant changes would be rather inconclusive without the knowledge gained from the sideband spectra. With this respect, the curves of Fig. 27.5 show that a fit of a spectrum with formal force constants, even if possible, may not reflect the physical situation. In fact, MACPHERSON and TIMuSK feel that a weakening of the core-shell force constant (by A g = - 200000

Vi 'c :> .ci .2 ~ :a 0 III .0 <t


K8r: KH ~ KBr : KH .- -- observed :: - calculated ':

, , , . , . , , ,

I a · · · · · · , · , ,

)J , , , , ,

\.\ 60 80 Olcm!

Frequency

KBr: KH

KBr: KH

50 Frequency

Fig. 27.5. (a) Induced far·infrared absorption due to H - ions in KBr. The calculated curve is from the shell-model defect with force-constant changes dj, = - 8 855 dyn/cm, dj2 = ~ 41 00 dyn/cm and d g = 0 dyn/cm which are the values obtained from fitting to the sideband spectrum. The main feature is the strong peak at 88.5 em - 1 which is attributed to a resonance of the perturbed T, u

modes of the lattice; (b) effect of dj, on the calculated far-infrared absorption in KBr:H -. The six plots are for dj, equal to - 5 855, - 6 855, -7855, - 8 855, -9855 and -10855 dyn/cm and dj2 =

-4100dyn/cm. The effect is to move the two peaks to lower frequencies as dj, becomes more negative. The peak heights and widths are also greatly affected. The heavy curve is for dj, =

b

c

d


dyn/cm) gives a more satisfying theoretical absorption spectrum. MACPHERSON and TIMUSK have studied the far-infrared absorption spectrum induced by Ucenters in various crystals.

The shell-model description requires a larger number of parameters than a rigid-ion model; in fact, these may not even be sufficient: additional electronic degrees of freedom may be important. The experimental data may not allow one to assign values to all of these parameters. Even if the experimental data can be described within a certain error limit by a few parameters, these may not reflect the experimental situation. Experiments of different kinds often reveal different and complementary information. It is hoped that the use of models of various kinds will lead to an understanding of the electronic properties of defects. After all, there are electrons and not springs between the ionic cores.

d) Shell-model interpretation of the effective charge. In lattice dynamics the effective charge (Sect. 4b) is a rather well-understood quantity. The bare charge of a vibrating ion is screened by the polarizing electrons. In unperturbed crystals the absorption of light by a Reststrahlen phonon is proportional to this effective charge. In a perturbed lattice, phonons with wave vector other than zero become infrared active.

From Sect. 25 b the dielectric susceptibility of the defective crystal is given by

(25.17')

with

The shell model will be used to interpret the charges, Green functions and defect matrices. For a perturbed crystal one has to work in the core and shell space on the one hand (as above) and in the impurity and rest space (as in Sect. 25) on the other. The unperturbed Green function tfio as defined in (27.9) consists of two parts, one dependent on and one independent of frequency,

tfio = tfjl + tfj2

with

- 8855 dyn/cm as in (a); (c) effect of Af2 on the calculated far-infrared absorption in KBr:H-. The six plots are for variations in Af2 from -3600dyn/cm to -6100dyn/cm in steps of -500dyn/cm and with Afl = - 8 855 dyn/cm. As .112 becomes more negative, the effect is a shift to lower frequencies of the two peaks. The high-frequency peak is also reduced in amplitude and broadened and the lower-frequency peak increases in strength. The heavy curve is as in (a); (d) effect of Ag on the calculated far-infrared absorption in KBr:H-. The plots are for Ag=O, ... , -500000 dyn/cm in increments of -100000 dyn/cm. As Ag becomes more negative, the peaks are shifted down in frequency, the high-frequency peak becomes weaker, and the low-frequency peak becomes stronger.

The heavy curve is for Ag=O (as in (a». (MACPHERSON and nMUSK, 1970b)


and

cf. (27.6). Similarly, for the perturbed Green function,

g=gL +gE, cf. (27.9). Also

Sect. 27

To demonstrate the idea of the effective charge, consider a crystal without defects. Then the dielectric susceptibility is given by

N T X~p(w) =-V zo,~ ~o(O) zo,p

= ~ z6.a ~lcdO) zo,p+X~p(oo) (27.12)

where (27.13)

and

(27.14)

Zo,~ is the effective charge as it usually appears for the q = 0 vibration. The 4J oPQ (O) are the Fourier transforms of 4J oPQ for q ~O which are defined analogously to ~ OPQ(O). e( 00) = 1 + 4n X"p( 00) is the high-frequency dielectric constant which arises from (virtual) electronic excitations and which can be shown to be given for the NaCllattice by the Clausius-Mosotti relation

IX e(00)=1+4n 4n

1-- IX 3

where IX is the electronic polarizability,

N T R -1 IX= V Zos(4Joss(O)) . zos'

The graphical representation of (27.12) is shown in Fig. 27.6. Two special cases for a crystal with a defect can be considered. First assume no

charge changes. This is typical for a defect whose charge and polarizability are similar to those of the replaced host atom, for example an isoelectronic defect in

X(w)= (0-0--- ... )-(0-.---0)+0----0

Fig. 27.6. Graphical representation of (27.12). The open circles denote Z, the coupling constant between the external light and the phonons or electrons. The other symbols are as in Fig. 27.2


an alkali halide. Then

XIlP(W) = XIlP( ex))+ ~ zL @ledO) zo,p

1 T -2 - V ZOSIl ~2SS(0) tSS~2SS(0) zosp + O(N )

(27.15)

with

. (tee tes') ( 1 ) ~ (0) tse tss - ~2SS(0) tPosdO) 1 ee ,

and XIlP(ex)) as in (27.14) above. Here the high-frequency susceptibility is essentially unchanged, first term in (27.12'); the second term shows that the effective charge as well as the dispersion oscillator is changed; the third term is usually very small (due to the factors ~2)'

On the other extreme, for a diamond lattice one has

Z~'1l ~1(0) zo,p=O,

and anyone-phonon absorption must be due to defect-induced charge changes; for a diamond lattice one has

XIlP(W) = XIlp( ex)) +~ L1 ZllgL L1 zp - ~ ZSIl ~2SS(0) tss ~2SS(0) zsp

with T 1 T

XIlP( ex)) = ZOSIl ~2SS(0) zosp + N L1 ZSIl gE L1 zSP'

ZSIl=ZOSIl+ ~ [(l-tgo) L1 ZIl]S'

In the general case it seems to make even less sense to carry out the matrix multiplications. One should go back to (25.17') and the equations following (25.17); the physical interpretation of the susceptibility is then along the lines of Sect. 25b and that of the core and shell elements as in Sect. 27b.

In the general case, the effective host as well as the (frequency-dependent) defect charge determines the absorption. It should be stressed that the shellmodel matrix formulation is the appropriate one except in cases where one might be interested in the effective charge connected with a particular (ideal) lattice mode.

The total integrated absorption in a perturbed crystal turns out to be proportional to, cf. (25.7),


with 7

Z,,=Zc,,- tPcs tPSSl Zs,,'

Differently from (25.7') for the rigid-ion case, the integrated absorption here does depend on changes in the force constants.

In the following section an anharmonic generalization of the foregoing arguments will be given.

e) The higher-order dipole moments. A formal interpretation of the secondorder dipole moment, based on symmetry arguments, is that a phonon perturbs the perfect structure of the lattice, thus allowing an otherwise forbidden phonon to be excited. Another interpretation is that the incoming light causes a virtual electronic excitation which decays into two phonons. This latter interpretation will be supported by the following considerations.

In an anharmonic crystal the dielectric susceptibility due to the first-order dipole moment is given by

(27.16)

where GA denotes the anharmonic one-phonon Green function. In terms of the harmonic Green function, GH, and the self-energy, ll, the anharmonic Green function is given by

The susceptibility above can be interpreted in terms of shell-model quantities. Then the harmonic perturbed Green function consists of two parts, one being dependent on the frequency, the other being a real constant,

(27.9)

where GE has non-zero elements in the shell subspaces only. The subscript E refers to the fact that GE is the remnant of an electronic Green function, as explained in Sect. 27 c.

To demonstrate the effect of GE one expands the anharmonic Green function with respect to GE • One obtains

with

GL =(l+GLllt 1 GL ,

ii=ll(l+G~ll)-l.

(27.17)

Inserted into (27.16), the term GE gives X"p(oo). To demonstrate the effect of the term GEllGE, II shall be assumed to be due to two-phonon decay processes. Then II is given by (see Sect. 30d)

(27.18)

Here tP3 is the third-order anharmonic coupling-constant tensor and G2 is the two-phonon Green-function tensor. Substituting expression (27.18), one obtains


a OC = 0- - - -e::::

b OE = 0-----eE

c <d- = o---~ Fig. 27.7 a-c. ~ Graphical representation of higher-order dipole moments. (a) Second-order dipole moment, (27.19); (b) third-order dipole moment; (c) contribution to the renormalized first-order

dipole moment

Upon comparison of this expression with that due to the second-order dipole moment, Z~2l,

LlX (w)=!~Z(2)TG Z(2) a/3 2 V a 2 /3'

one can see that the second-order dipole moment, Z~2), can be interpreted as

(27.19)

In a graphical representation (27.19) is given in Fig. 27.7a. Recall that Za has elements in the core and shell (but, for example, not breathing for the breathing shell model - BSM) subspaces and that GE has non-zero elements only in the shell (and possibly breathing in the BSM) but not core subspaces. Also remember that GE describes virtual electronic transitions. Then it is seen that the second-order dipole moment can be interpreted exactly as stated above. In the same manner, higher-order dipole moments as well as renormalized dipole moments can be interpreted with the help of higher-order decay processes and those terms of the self-energy which are not frequency dependent (and, therefore, real only), respectively, see as an example Fig. 27.7b and c.

The relation between anharmonicity and higher-order dipole moments was first pointed out by KEATING and RUPPRECHT (1965) and KEATING (1965). OITMAA (1967) gave a perturbation-theoretic shell-model description of higherorder dipole moments. For the relation of the second-order dipole moment to the susceptibility change see BELLOMONTE (1977b).

Again, one should stress here, that the shell-model formulation of the dielectric susceptibility in tensor notation, together with expressions (27.16) and (27.9), automatically takes care of Xa/3(!Xl) (and its modification due to electronphonon interaction) as well as of higher-order dipole moments. The expansion in terms of GE has been done here solely for illustration.

f) Shell-model interpretation of the Raman scattering intensity. This section presents a short microscopic interpretation of the polarizability and its derivatives as they appear in the expression for the Raman scattering tensor, see for example (25.31). After the derivation of the usual electronic (non-resonance) Raman scattering tensor the extensions towards the ionic and the resonance Raman effect will be indicated.


One starts out from the electronic polarizability, actually, the electronic susceptibility, Xa/3( (0) = X NjV, which is given by (27.14),

(27.14')

(neglecting the tensor aspect of Z and thus X). Here 7 Zs is the linear dipole moment which couples the electronic excitation to the electro-magnetic radiation; cPs;, 1 is the remnant of the electronic propagator in the Born-Oppenheimer approximation. If the crystal and/or the defect have electronic degrees of freedom other than the dipolar one (index S), i.e. breathing or tetragonal deformations (index B), then the electronic propagator is given by 7

cPs;,l =(cP~~)-cPSB cPi.li cPBS)-l,

where the index (0) refers to the bare electronic dipole-dipole interaction. Expanding the electronic susceptibility in terms of ionic core, shell and other

deformation displacements to first order one obtains the susceptibility fluctuations as the first-order expansion term (neglecting dZjdu),

X=Z~ cPSS 1 Zs+Z~c.L: updcPssljdup] Zs p

X=Z~ cPs;, 1 Zs - Z~ cPs;, 1 [L: (cPSPS - cPSPB cPi.li cP BS - cPSB cPi.li cP BPS p ~----

+ cPSB cPi.li cP BPB cPi.l~ cP BS) Up] cPs;,l Zs (27.20) ~~~

or

x = Z~ cPs;, 1 Zs - Z~ cPs;, 1 [L: cPSPS Up] cPs;, 1 ZS. (27.21) P

Here again, one has defined an effective shell-shell anharmonicity, cPsps , which contains other deformation degrees of freedom besides the bare dipolar degrees of freedom which by themselves result in just cPsps ,

dcPss/dup == cPsPs = cPsps - cPSPB cPi.l~ cP BS - cPSB cPi.l~ cP BPS

+ cPSB cPi.li cP BPB cPi.li cP BS ·

Equation (27.20) directly gives the first-order polarizability p~l) as 7

(27.22)

(27.23)

in terms of shell-model quantities, thereby allowing the following physical interpretation of the Raman scattering process: The light causes via the linear electronic dipole moment (Zs) a virtual electronic excitation which propagates (cPs;, 1 ) through the crystal. Due to electron-ion interaction (cPsps) a phonon is absorbed or emitted, and another electronic excitation propagates until it finally creates a light wave again.

Furthermore, the shell-model interpretation shows that the shell model provides an effective frequency-dependent first-order polarizability (similar to the effective charge, see Sect. 27 d above), which is obtained from the Raman


tensor as proportional to

(27.24)

after elimination of the shell and breathing degrees of freedom. It should be noted here that COWLEY (1964) uses a different method to obtain the higherorder polarizabilities. In essence he employs the adiabatic condition for the lattice potential and finds the polarizabilities by the method of equal powers. COWLEY's method leads, of course, to the same results, but is enormously labourious compared with the relatively simple derivation in (27.20). The connection between polarizability expansion terms and lattice anharmonicity was first pointed out by LOUDON (1964b, 1965).

In a simple (not breathing) shell model the scattering intensity in this description is governed by the anharmonicities. Assuming only one polarizable sublattice and a strong shell-shell anharmonicity between the defect at a halide site and its nearest neighbors in a NaCl-structure crystal the resulting expression for the first-order polarizability has a form like in Sect. 25c where now a, b, and c are to a good approximation given by

4isss (O, x; 0, x; 100, x), 4isss (O, x; 0, x; 010, y), 4isss (O, x; 0, y; 100, y)

times [Zs(O) 4iss (O, x; 0, X)]2. In this same approximation the scattering intensity for a defect at an alkali site would be zero. 8 From Born-Mayer and Coulomb potentials, typical for alkali halides, one would expect that b is equal to c and that a and b have opposite sign. This means that the ~i spectrum would be stronger than the ~+ and r;~ spectra. The scattering intensity here is directly related to anharmonic properties of the defect.

Compared with the expression for a simple shell model (without breathing), expression (27.24) contains additional terms. These terms represent extra scattering intensity. If one assumes, for example, that the defect has a dominating breathing degree of freedom, then the correspondingly strong contribution from

to the spectrum of the scattered light has breathing (1;+, A1g) symmetry. This picture seems to be supported by Raman measurements by MOLLER and KAISER (1972, KAISER and MOLLER, 1969). They find groups of systems. Substitution of an alkali ion (Li+) in alkali halides results in ~+ spectra which are weaker than the r;r spectra as predicted above. Substitution of halide ions reveals r;+, r;r and Tzt spectra; however, due to the breathing degree of freedom of the halide defects, the ~+ spectra are stronger than the other spectra by a factor of about 5. Substitution of Tl + (HARLEY et aI., 1971) and Ag+ ions, finally, gives strong r;r, much weaker Tzt and essentially no r;+ spectra. Comparing this result with the one obtained from the Li + defect, one is tempted to conclude

8 This is most easily seen if core and shell displacements are reinterpreted as ionic and polarization displacements; then the alkali sublattice does not have any polarization degrees of freedom.


that tetragonal electronic degrees fo freedom (originating from d electrons) are responsible for the strong r;i and Tz~ intensities.

For the higher-order Raman effect one has to proceed a little carefully. Upon expanding x( CXl) to the second order one obtains the following two elements in addition to the terms in (27.21)

t p(2) U U = Z~ CPs;, 1 (I CPsPS Up) CPs;, 1 (I CPSQS UQ) CPs;, 1 Zs P Q

+ Z~ CPs;, 1 (I CPSPB Up) CPi3i (I CPBQS U Q) CPs;, 1 Zs P Q

- Z~ CPs;, 1 (t I CPSPQS Up U Q) CPs;, 1 Zs PQ

where CPSPB and CPsPQS are defined analogously to CPsPs as given by (27.22). In a graphical representation the expansion of x( (0) to second order (neglect

ing the effect of breathing) is given by diagrams a) to d) of Fig. 27.8, and the corresponding first- and second-order Raman scattering intensity is shown graphically in parts a) to c) of Fig. 27.9. There is also an interference term between the processes b) and c). The Raman scattering process as shown in Fig. 27.9d is missing so far from our analysis; one would expect this process to arise from a second-order susceptibility contribution, as shown in Fig. 27.8e. This process is easily obtained, if the harmonic phonon propagator in Fig. 27.9a is replaced by the anharmonic one. The desired process then results from twophonon decay in the self-energy.

The expansion (27.20) or (27.21) has been made in terms of ionic displacements. In principle, the expansion can also be made in terms of electronic displacements. In the adiabatic approximation, where the electronic propagator is purely real, no electronic absorption processes can occur. However, if electron-phonon interaction is included for the electronic propagator, a scattering process as is shown in Fig. 27.ge becomes possible, and there is no reason why this process should not occur. There are interference processes between all processes which are shown as parts b) to c) in Fig. 27.9.

On the other hand, by expanding the total, ionic and electronic, polarizability, phonon propagators show up, too, wherever electron propagators occur in

a X (00) = 0-----0

b

c,d + O----~--~---<> - O---~---<>

e, f o----.----{) A

Fig.27.8a-f. Graphical representation of the expansion of the electronic susceptibility in terms of phonon displacements (a) zero-order term; (b) first-order term; (c and d) second-order terms; (e) first-order expansion in terms of anharmonic phonons; (f) expansion in terms of electronic

displacements


e .............0- -~ - -0---.--0- - *- --0-----".~-"

Fig. 27.9 a-e. Graphical representation of the Raman scattering intensity. (a) First-order spectrum in terms of first-order polarizability (Fig. 27.8b); (b) second-order spectrum in terms of first-order polarizability (Fig. 27.8c); (c) second-order spectrum in terms of second-order polarizability (Fig. 27.8d); (d) first-order anharmonic phonon spectrum (Fig. 27.8e); (e) contribution from the po-

larizability expansion in terms of electronic displacements (Fig. 27.8 f)

Fig. 27.7 and 27.8. Thus the ionic Raman effect is included, as recently discussed (see references in Sect. 2Sc). Even though the frequency factor 0)4 of the external light in the expression for the scattered light intensity does not favor detection of a signal, it has to be pointed out that a localized mode or a (high-frequency) internal molecular vibration may give rise an ionic resonance Raman effect. Strong anharmonicity gives a strong first-order polarizability; on the other hand, a sharp resonance seems to require a weak anharmonicity.

A resonance in the electronic Raman effect is obtained from a shell model if the electronic mass is kept finite. Then the (bare) electronic propagator is

(<<P~~) - 0)2 MS)-l

which has poles around the frequency yCPss/Ms, corresponding to an electronic excitation energy.

Finally one should stress that a compact shell-model formulation, as indicated in Sect. 27 e, may be the appropriate representation. In a formalism like this, one would expand the total dipole moment (for higher-order dipole moments, Sect. 27 a) and the total susceptibility (for the Raman effect, this section) in terms of both the ionic and electronic displacements. Anharmonicity has then to be taken into account for both types. In the self-energy of the corresponding anharmonic propagators both types of degrees of freedom have again to be included. In the Born-Oppenheimer approximation all real electronic transitions are eliminated for which the shell model is not expected to account.


28. Concentration effects a) Introduction: Diagrammatic expansion. Up to here it has usually been

assumed that there is only one defect in the crystal. For actual systems this is not true. In this section the theory will be modified in order to include the effects of a finite (though small) number of defects.

Theoretically, consideration of a finite number of defects has the consequence that the non-zero elements of the perturbation matrix V, (22.8), are no longer restricted to a very small "impurity space". For impurity concentrations of, say, one percent, on the average every hundredth diagonal element of V refers to a defect. Nevertheless, most of the elements of V will be zero.

A simple way of handling a finite number of defects mathematically would be to distribute them in some way periodically over the crystal. However, an unphysical superlattice would be created in this way.

Expression (25.17) for the dielectric susceptibility of a crystal with one defect contains defect-induced terms which are down by a factor of liN compared with the host-lattice contribution. The ratio liN is just the fraction of defects in the sublattice under consideration. To account for a finite yet low concentration of defects, one would, therefore, try to replace liN by the concentration, p, of the defects. This will, in fact, turn out to be an approximation of the results below.

There are two basically different ways of approaching a many-defect crystal. The first approach is computer-oriented (DEAN, 1959, 1960, 1961, 1972; DEAN and BACON, 1965; PAYTON and VISSCHER, 1967a, b, 1968; ROSENSTOCK and MCGILL, 1968): a model lattice is assumed; defects in a given concentration are distributed over the crystal at random; the computer then finds the phonon eigenfrequencies and thus the density of states. With a knowledge of the eigenvectors, infrared and Raman spectra can be calculated in principle.

The other approach is analytical. Shift and width of a pure-lattice phonon, for example, are calculated for an average defect configuration. The averaging procedure is essential to this approach and various means of performing it have been suggested.

A rather simple way to treat the randomly disturbed crystal is the virtualcrystal approximation (NORDHEIM, 1931; EDWARDS, 1962) in which the "scattering potential", V, is replaced by its average, <V), which is distributed over the total sublattice, thereby restoring a translationally invariant crystal. This model is expected to work well if < V) and V are not much different from one another.

If the density of defects is low enough to allow multiple scattering to be neglected so that only single scattering has to be considered, the optical model is expected to be appropriate (FOLDY, 1945; LEVINE and SANDERS, 1967). Application of this approximation to Green functions has been developed by LANGER (1961); MATSUBARA and TOYOZAWA (1961); and TAKENO (1962a). This method along with others has been reviewed by MARADUDIN (1966a), and will be presented in Sect. 28 b.

Self-consistent optical-model-type approximations have been suggested by DAVIES and LANGER (1963); YONEZAWA (1964); LEATH (1968); and will be mentioned in Sect. 28 c.

The coherent potential approximation (SOVEN, 1967) is a combination of the virtual-crystal model and the optical model. Here the phonons of an average

Sect. 28 Concentration effects 383

(" coherent") lattice are scattered off defect as well as host-lattice sites. The method has been applied to electrons (YONEZAWA, 1968; VELICKY et al., 1968; KIRKPATRICK et aI., 1969), to excitons (ONODERA and TOYOZAWA, 1968), to phonons (TAYLOR, 1967; TAKENO, 1968b), and to magnons (EDWARDS and JONES, 1971a, b; BUYERS et aI., 1972, 1973; ELLIOTT and PEPPER, 1973; HARRIS et aI., 1974), and will be outlined in Sect. 28d. Much of the work in the coherent potential and other self-consistent approximations has its origin in the multiple-scattering method of LAX (1951). The coherent potential approximation is today considered to be the best approximation for finite concentrations. Even though many of the different methods and approximations can easily be discussed in terms of simplifications of the CPA (FREED and COHEN, 1971), here simpler methods will be developed first in order to give an easier introduction to the language and concepts of lattices with finite concentrations of defects. Still another theoretical approach is that by continued fractions, see e.g. KELLY (1980).

There are also a number of assumptions and approximations connected with most of the optical-model and coherent-potential treatments, most importantly that of randomness of the defect distribution. This seems to be somewhat against the results of Sect. 22b where it was found that the energy (in the harmonic approximation) due to a defect changes by an amount

with

LlE1 =4)1S+!S4)2S = -!4)1 GST 4)1

S= _GST 4)1.

Here all quantities refer to a single defect. The energy lowering due to a defect pair, LlE2, will usually be different from the change due to two single defects, 2L1E1, resulting in al) energy difference, LlE2-2L1E1, and in non-randomness if this is negative. On the other hand, if the energy difference is much less than the energy changes, the assumption of random statistical distribution will be a good approximation. Defect correlations have been considered by HARTMANN (1968).

Another approximation, treating the defect as an isotopic one, has similar grounds. Accidental close neighbors may have overlapping impurity spaces, and the sum of the elements in the common subspace may possibly be different from those of the defect pair. This becomes most obvious for two neighboring vacancies in a crystal with nearest-neighbor interaction (-f) only: Let the change LI ([> in force constants between the vacancy and its neighbors be f Two neighboring vacancies would also have a change f between them while two single-defect matrices would result in a change of 2f in the subspace of both matrices. However, this is a very extreme example, and in most cases it may be a good approximation to let the single-defect matrices simply add (BEHERA and DEO, 1967; BEHERA and TRIPATHI, 1974; TRIPATHI and BEHERA, 1974; LAKATOS and KRUMHANSL, 1968, 1969; KAPLAN and MOSTOLLER, 1974a; FuKUYAMA et aI., 1974). Nevertheless, much of the theoretical work which has been done so far has assumed isotopic defects. In this connection the "model" Lifshitz formalism of Sect. 27 is very useful, if not necessary, because it reduces


the change in the long-range Coulomb interaction between two defects to the respective change at two isolated defects.

The perturbation due to a defect is also assumed to be the same for every defect. A random variation of the change in force constants (DYSON, 1953; TAKENO and GODA, 1972; and references therein) seems more applicable to amorphous substances which we do not wish to deal with here. A review on amorphous materials has been given by BOTTGER (1974) and CONNELL (1975), see also the conference proceedings edited by LUCOVSKY and GALEENER (1976).

The final, more crucial approximation has been made for reasons of numerical evaluation. Green functions for actual three-dimensional crystals are rather complicated, whereas those of simple linear chains can be handled analytically to a certain extent. Linear chains, however, often represent academic exercises rather than physical relevance.

The following sections will concentrate on the effects of "single-site" scattering, by which is meant the neglect of all processes in which a phonon is scattered off clusters which consist of more than a single defect. This limits the discussion to relatively low impurity concentrations for which the probability of two defects forming a pair is relatively small. Scattering by many pairs has been considered by a number of authors (LANGER, 1961; YONEZAWA and MATSUBARA, 1966b; LEATH and GOODMAN, 1968; SOVEN, 1969; AIYER et a!., 1969; NICKEL and KRUMHANSL, 1971a, b; CYROT-LACKMANN and DUCASTELLE, 1971; SCHWARTZ and SIGGIA, 1972; SCHWARTZ and EHRENREICH, 1972; LEATH, 1972; BRENIG and SCHONHAMMER, 1973; MAURO and SCARFONE, 1973) with slightly different results. The problem in treating pairs or larger clusters (TAKENO, 1969; CAPEK, 1971; FREED and COHEN, 1971; BUTLER and KOHN, 1970; BUTLER, 1972, 1973; NICKEL and BUTLER, 1973; HEHL, 1972; Foo et a!., 1973; KOZYRENKO and MIKHAILOV, 1973; LEATH, 1973; MILLS, 1973; NnzAKI, 1973; ZITTARTZ, 1974a, b) is to find the correct treatment for multiple occupancy of a lattice site by defects. The work done so far has mainly used diagrammatic or cluster expansions. A conceptually simple differential method has been proposed by PERSHAN and LACINA (1968). An algebraic method has been developed by KLVANA (1970) and BRUNO and TAYLOR (1971), and various other schemes have been discussed (BROUERS et a!., 1973; BISHOP and MOOKERJEE, 1974; DALLACASA, 1974; STERN and ZIN, 1974; BRENIG and MARTIN, 1975; LLOYD and BEST, 1975).

A comprehensive review of methods and applications connected with large defect concentrations has been given by ELLIOTT et a!. (1974), ELLIOTT and LEATH (1975), and BARKER and SIEVERS (1975).

The effects of defect pairs are roughly proportional to p2 and even at low concentrations are important near band edges. Reference is made to the detailed work by LIFSHITZ (1964).

Even though the Green function as such cannot be observed experimentally, one applies the different methods to the Green function to begin with in order to avoid unnecessary complications, only afterwards applications to the dielectric susceptibility (ELLIOTT and TAYLOR, 1967) are made, see also Sect. 31.

From Sect. 23 the Green function of the perturbed crystal is G=Gn-Gn VGn+Gn V(;_ V(;_ - 4-


where V now contains the changes due to all defects. Taking a statistical average over all defect configurations, one obtains the average Green function

(G)=Go-Go(V) Go+Go(VGo V) Go- + .... (28.1)

Denote the perturbation matrix connected with the defect at site xm by

1 2 ... '; ...

The perturbation is taken to be the same for every defect. The Greek indices will be restricted to refer only to defects while a Latin index may refer to any lattice particle. For example, for the first-order expansion term of (G) one obtains

[Go(V)GoJi=LGh~(V~)Gi/' (28.2) ~

The probability of finding a defect at any arbitrary site is equal to the concentration of defects, p. One therefore finds from (28.2)

[Go (V) GoJij = p L Ghk V k Gii k

= p L Gt v Gii (28.3) k

where the column of Gt and the rows of Gii are now within the impurity space of a defect at site X (Lk). Note that v is a matrix and that the Greek index ~ in (28.2) refers to the impurity space of various defects in a given configuration. Finally, the index k refers to an impurity at any possible site in the crystal. In components I, K, IX one has

[Go(V) GOJlrca;I'rc'a'

where the indices Ik etc. refer to the sites of a defect matrix of a defect centered at site X(lk' Kk).

Analogously, as another example, one obtains for the third-order term of (G)

[Go(VGo VGo V) GoJi= LGh~(V~Gb'V'Gb~V~)G1/ ~,~

= "GikvGkmvGmnvGni{p3(1_J )(1-15 )(1-15 ) L... 0 0 0 0 km mn nk kmn

The three terms in curly brackets denote scattering of a phonon at three different defects, two different defects, and a single defect, the probability of their occurrence being given by p3, p2, and p, respectively.

386

-x-*-x-, , , , , , , , , , , , •••


, [ -x-x-x- -x-x-x- -x-*-x- J

+ ! \ / + \,1', + \, ,/ + . " " . ~

~X-)E-X-, ' I , , I

, , I

'e'

Sect. 28

Fig. 28.1. Graphical representation of expression (28.4). The three terms represent scattering at three different defects, two defects, and one defect, respectively. Full line G; cross V'; dot x(O

The different scattering processes of the third-order term can be represented by graphs, introduced by LANGER (1961), as in Fig. 28.1. Each part of the solid lines represents the phonon propagator, Go, the dots represent a defect site and the crosses represent a scattering matrix, v. The niles for incorporating summations, powers of the concentration, and Kronecker delta functions vary from approximation to approximation, since the complicated summations as in (28.4) cannot be performed exactly.

The form of the first diagram in Fig. 28.1, which looks like a repetition of a simpler one with one dashed line as given by (28.3), suggests that one looks for an approximation in the form of a Dyson equation

<G)=Go-GoIGo+GoIGoIGo- + ... =Go-GoI<G) =[Go1+I]-1. (28.5)

If one represents <G) by a thick line, Go by a thin line, and I by a black square, (28.5) can be represented by graphs as in Fig. 28.2a. In I one has to include all diagrams appearing in the expansion of <G), (28.1), which cannot be separated into two parts by cutting a single-phonon line. Some of these terms are shown in Fig. 28.2 b. I can be regarded as a self-energy localized at all lattice sites. Since it is a harmonic quantity, it is temperature-independent. Its real part may be understood as an average phonon energy change due to statistical perturbations of the lattice, and the imaginary part as an inverse phonon lifetime due to scattering of the lattice phonons at the defects.

b) Low-concentration single-site scattering approximation. The first three terms of the self-energy as represented by the graphs of Fig. 28.2b describe

a _ • + •• - + ...

•

b •

• • f', , 1 1 1 I \ 1 1 I ,

X x-x x-x-x x-x-x x-x-x-x x-x-x-x 1 I I \ I I \ I \ 1/ \ / + - ... I I I + \1/ - + ... + \ / \ 1/ \ I 1 II \II 'fI 'j 'fI • • •

Fig. 28.2a, b. Graphical representation of (a) the Dyson equation for a perturbed lattice, (28.5); and (b) the harmonic self-energy. Thick line, <G); black square, 2:; other symbols, see Fig. 28.1

Sect. 28

• ~ · • · •

Concentration effects

~ >f-);< I _ \,'

I " • • x-x-x

+ \! ,/ -'" e

387

~ + ... = :

x-x \ . \/ • .. •

Fig. 28.3. Graphical representation of the single-site scattering approximation for the harmonic selfenergy

(repeated) scattering of a phonon at the same defect, while the other three graphs represent scattering at a defect pair. One is neglecting these and more complicated processes in the simplest, single-site scattering, approximation for the self-energy. Writing out all terms of (28.1) as in (28.4), discarding all terms with graphs whose broken lines cross (when the dots are drawn on the same side of the unperturbed propagator line) like the ones represented by the last three graphs of Fig. 28.2b, and regrouping the Kronecker delta functions, one obtains from (28.4) and (28.5) a single-defect scattering approximation for the harmonic self-energy

with /T(1) = P V - p(l- p) v go v + p(l- p)2 v(go v)2 - + ...

=pv[l+(l-p)gov]-l

=pt(l), (28.6)

represented in Fig. 28.3. For the Green function one has in this approximation

<G) =(G0 1 + .1')-1

(28.7)

Here again, a notation is used like in the Lifshitz formalism, and matrices in an impurity subspace are denoted by lower-case letters. After averaging, the Green function has become translation ally invariant again, since now each site (within one sublattice) is occupied by a defect with probability p.

Compared with the pure-lattice Green function, the poles of the perturbed one are shifted. To first order the effects are proportional to the defect concentration, p; shift and width are given in terms of tel), a quantity very similar to the t matrix, (23.6). One may now argue that the theory, being restricted to single-defect scattering, is valid within the limit p ~ 1 only, so that all factors (1- p) in (28.6) can be replaced to a good approximation by 1. Then one has

.1'ij~/T(l)(jij' /T(1)=pt, t=v(l+goV)-l. (28.6')

To first order in p one finds for the average Green function Go, (28.7),

which is written short as

Go~Go-pGotGo·

This is the intuitive generalization of the result (23.8 c), but more restrictive than the approximations (28.6) or (28.6').


c) Self-consistent approximation. According to the theory above, the Green function at a local- or gap-mode frequency exhibits a pole in the form of a delta function. This is due to the fact that go and thus 0.(1), (28.6'), is purely real outside the phonon bands. On the other hand, defect-defect interaction leads to a broadening of the resonance line. The perturbed Green function does have a non-zero imaginary part at the resonance frequency. The idea therefore is to substitute the average perturbed Green function, go, for the unperturbed one, go, in the expression for the self-energy, (28.6) or (28.6'), and to solve for go selfconsistently,

(28.8) or possibly

(28.9)

This procedure allows processes represented by "nested" diagrams as in Fig. 28.4, or like the fourth and fifth terms in Fig. 28.2b, to be included. However, it seems rather awkward to handle the dependence on the defect concentration in a correct way. This is due to the fact that in the framework of the simple approximation leading to (28.6) the defects appearing in the nested diagrams can only be defects at different sites.

Even though both approximations, (28.6) and (28.8), give the correct results

a(l)=O for p=O

a(l)=v for p= 1

the slope aGo/a p is incompatible for concentrations p around zero and around one (YONEZAwA and MATSUBARA, 1966a; for example). This again says that the approximations above are valid only for low concentrations in which case the factor 1 - p can be approximated by 1.

d) Coherent potential approximation (CPA). In order to extend the range of validity for a single-defect approximation to at least very low and very high concentrations (and hopefully to intermediate concentrations as well), the coherent potential approximation has been proposed. In this approximation a virtual crystal is constructed by an average ("coherent") potential change w at every site of the sublattice which contains the defect. A phonon then is scattered off a potential v - w with probability p, and off a potential - w with probability (1- p). The coherent potential is' determined by the fact that on the one hand one defines

~ I,

Ie' , I , , I , , , ,

-x-x-~

Fig. 28.4. Examples of processes which are neglected in the single-site scattering approximation


and on the other one has

G=(G0 1 + V)-l =[G0 1 + W+(V- W)]-l

=(G0 1 + W)-1_(G0 1+ W)-l(V- W)[1+(Go1 + W)-l(V- W)]-l(GOl +W)-l

=<G)-<G) T<G).

Taking the average of G one has

<T)=«V- W)[l+<G)(V- W)]-l)=O

by which W is defined. In the single-defect scattering approximation < G) is given by

<G) ~Go =(G0 1 +wl)-l,

and (28.10) reduces to

where now C<l)=p, t(l)=V(l)(l+goV(l»-l, V(l)=V-W,

c(2)=1-p, t(2)=v(2l(1+goV(2»-1, V(2)= -w.

(28.10)

(28.11)

(28.12)

(28.13 a)

(28.13 b)

For practical calculations it has to be assumed that w has short-range components, i.e. is a matrix of small dimension. The larger of the dimensions of w and v then determines the dimensionality of t(2). From (28.12) one obtains

w=pv[l+go(V-W)]-l. (28.14)

To lowest order in p one finds

w~Pt=pv(l+goV)-l=(1(l) for p~1

with t given by (28.9), and to lowest order in q = 1-p one has

w~v-(1-p)v(1+goV)-1 for 1-p~1

with the required symmetry for low and high concentrations.

e) Applications. As mentioned above, the Green function cannot be observed experimentally. Theories for the infrared absorption by crystals with many defects have been presented by HARDY (1964); ERNST (1967); ELLIOTT and TAYLOR (1967); TAKENO (1967, 1968a); and HARTMANN (1968) and a shell model formulation has been given by STRAUCH (1968). Far-infrared absorption measurements have revealed critical points in the spectrum (KLEIN and MAcDONALD, 1968; WOLL et aI., 1968), which shift to higher frequencies with increasing concentration (TIMUSK and WARD, 1969 ; WARD and TIMUSK, 1970, 1972). BRUNO et aI. (1970) have made a theoretical study of this shift; however, they detected an error in their computer program (BRUNO, private communication). HARRINGTON and WALKER (1970) have found a series of critical points in the sideband spectrum of the U.:center local mode in BaF2 which erode with increasing concentration of U-centers. They have also (1971) measured the width of the U-center local-mode line in BaF2 and find a co.s behavior, as predicted


from simple virtual-crystal-type assumptions (DAWBER and ELLIOTT, 1963 b); KIRBY et aI. (1968b) find a CO. 12 behavior for the KBr:Li+ low-frequency resonance; SHITIKOV et aI. (1980), for example, find a linear dependence of the resonance frequence and width on concentration in the Cu-Be alloy. No concentration dependence of the Raman-line frequencies and widths induced by Ni2+ ions and vacancies in LiCI was found by BATES et aI. (1977).

A more direct approach to shift and width of lattice phonons due to a finite concentration of defects may be made by inelastic-neutron-scattering techniques, but these have been carried out mainly for metal alloys. In most of these cases consideration of force-constant changes seems to be necessary to obtain agreement between theory and experiment. Some references are ELLIOTT and MARADUDIN (1965); NICKLOW et aI. (1968); BRUNO and TAYLOR (1971); STOCKS et aI. (1971); SVENSSON and KAMITAKAHARA (1971); COHEN and GILAT (1972); COHEN (1972); KESCHARWANI and AGRAWAL (1972, 1973); GAMBETTI et al. (1974); KUNITOMI et aI. (1973); KAMITAKAHARA and TAYLOR (1974); KAMITAKAHARA and BROCKHOUSE (1974); MOSTOLLER et aI. (1974); SACCHETTI and SPINELLI (1975). The resonance vibration (pole of a) of Ag defects in Al leads to a splitting of the frequency w(q) (pole of <G») at the resonance vibration (ZINKEN et aI., 1977). (KCI)1_JNH4CI)x has been studied by KAPLAN

5

4

~ 3 )(

G,2

1

0

a

b

£=-2.1 (~\ p= 0.25

l\·,1 \ !\ ;\i \.1\ I" -, y \

//1

0.2 0.4 0.6 O.B 1.0 x

£=-2.1 p= 0.25

0.4

0.0 f---+"'<::--=-I

)(

1~-0.4 a:::

-0.8

1.0 )(

1141

.§ 0.5

o 0.5 x

5 £=-2.1 4 p=0.75

)( 3 C).,2

0 0.2 0.4 0.6 O.B 1.0 x

£=-2.1

2 p= 0.75

Of----Io:::----;

Fig. 28.5. (a) Density of state p(x) (solid line) and g"(x) (dashed line) for the imperfect crystal with B=Am/mo= -2.1 for p=0.25 and p=O.75, as a function of the reduced frequency x=w/wmax ' W;" is the maximum frequency for p=1, (b) behavior of ii=W/w2mo as a function of x for e= -2.1 for

p=O.25 and p=O.75. (TAYLOR, 1967)


3.---------------------------, c=-2.1,p=0.1

2

0 0 0.2 0.4 0.6 0.8 1.0

x

4 (5=- 2, P=0.9

~

I I

3 I ~ I

I 11

~ I 11 I II

0,.2 I II I Ii

V nil I 1111 I 1111 I 1111 I Iv I I I I

I I I I I I

0 I

0 0.2 0.4 0.6 0.8 1.0 x

Fig.28.6. Comparison between CPA (solid line) and first-order (dashed line) densities of states for 8= -2.1 and p=O.lO and 0.90. (TAYLOR, 1967)

and MOSTOLLER (1974b, c). Neutron scattering is not considered in this chapter any further except for a paper by TAYLOR (1967) for illustration of the CPA. SEN and LUCOVSKY (1975) have compared the ATA and CPA for the phonon spectrum of a linear chain with isotopic substituents.

For the applications to the thermal conductivity of perturbed crystals, see e.g., SHARMA and BAHADUR (1975) or ALTUKHOV and ZAVT (1977). The Berreman effect for local modes has been investigated by ZAVT (1963), DOETSCH (1969b), and MARADUDIN and OITMAA (1969).

TAYLOR (1967) has done a rather extensive study of the CU1_pAup alloy treating the Au as a substitutional isotopic defect (8=L1mjm= -2.1). TAYLOR'S CP A results for the Green functions and the related densities of states are given in Fig. 28.5 a together with the coherent potential, w, in Fig. 28.5 b. The coherent potential shows a reasonance behaviour, and even may show divergences. The density of states for the CPA (28.11) and the simple single-site scattering result, (28.7), are compared for very low and high concentrations in Fig. 28.6; they show very close agreement except in the local-mode frequency region. A comparison of TAYLOR'S CPA results with the computer calculations of PAYTON and VISSCHER (1967a) is made in Fig. 28.7. The machine calculations show definitely more detailed structure in the local-mode frequency


2.5.----------------, 5r-------------,

2.0

~ 1.5

5

4

to< 3

Qr 2

( i l t' = - 2, P = 0.240

(iii It'=-2,p=0.760

4

5

4

0.8

(i i l t' =-2, p=0.491

(iv l 6'=-2, p=0.866

0.2 0.4 0.6 0.8 to

Fig.28.7. Comparison of CPA densities of states (line) with the machine calculations of PAYTON and VISSCHER (1967a) (histogram) for a simple cubic crystal: 8= -2, (i) p=0.240, (ii) p=0.491,

(iii) p=O.760, (iv) p=O.866. (TAYLOR, 1967)

region, which is attributed to defect clusters. This cannot be reproduced by a single-site scattering theory. Otherwise the agreement is excellent.

In calculating the density of states in Sect. 25a only the diagonal terms of the perturbed Green function have been used. The off-diagonal elements are also needed for the calculation of the dielectric susceptibility, see Sect. 25b,

where Za(L, y) is the charge tensor connected with the charge on the particle at site x(L). Performing the average over defect configurations one obtains

1 T <Xap(w) = V <Za G Zp), (28.15)

one can proceed as in Sect. 28b and collect terms which are multiplied from the left and the right by the unperturbed charge and by the charge perturbation. Another possibility might be not to distinguish between unperturbed charges and charge perturbations but instead between host and defect charges. However, the analysis of (25.6) and (25.16) shows that the former alternative is the correct wav to nroceed.


Neglecting all but single-defect scattering processes, as in Sect. 28b, one obtains a susceptibility

1 -(X<xp(w) ~ X"p(w) = V {zL ~o(O) Zo,p

+pLl z!'[1-(1-p)got] ~0(0) zo,p T -+ pZo,,, ~0(0) [1-(1- p)t go] LI zp

+ p2 LI z; [1-(1- p)got] ~0(0) [1-(1- p)tgo] LI zp

+ p(1- p)LI z; [go -(1- p)got go] LI zp}. (28.16)

Since the approximation is valid for low concentrations only, one replaces 1-p by 1 and obtains

with %,,= zo" + p[1- tgo] LI z"

~0(0)= {[~0(0)] -1 + pt} -1,

(28.17)

(28.18)

(28.19)

in complete analogy to (25.17) to (25.19). Equation (28.17) is graphically represented in Fig. 28.8 and can be interpreted as in Sect. 25b: The light couples to all (unperturbed) charges Zo and excites the modified Reststrahlen oscillator (<@0(0)). The probability for excitation of a phonon by the light via a charge perturbation at a defect site is proportional to p. The phonon, once excited, either emerges directly into the random lattice (LlzTl) or is first scattered at the respective defect (LI ZT go t) and then propagates through the random lattice (~(O)). It is finally annihilated again at unperturbed charges and, with probability p, at some other defect with or without scattering. The phonon also may be excited and annihilated at some defect with probability p with or without having been scattered at this defect, but without having emerged into the random lattice (second term of (28.17)).

Replacing go by go in (28.17) and (28.18) gives a self-consistent single-defect scattering approximation for (28.15).

In the coherent potential approximation things are slightly more complicated, since a phonon is now scattered off every defect as well as every hostlattice site. The average (homogeneous) charge is

(28.20)

the charge perturbations are

(28.21)

t ' ft] [ , " ] - I /. 1-, X Ol(l (w) = 0 + I - /: - 0 + I - 5', +

6 O-~ 6 ~---b Fig. 28.8. Graphical representation of the expression for the susceptibility, (28.17)


The physical content of (28.17) is not changed. One must have, therefore, in the CPA with (28.20) and (28.13)

with

XaP (OJ) = Z; <!io (0) Z P + L eli) ,1 z~)T g<i) ,1 z~) i

Za= Za + L c(i)(l- t(i) go),1 Z~) i

g(i)=go- go t(i) go=(l- go V(i»)-l go

t(i) = v(i)(l- go V(i»)-l.

In the low-concentration limit one finds

(28.22)

and one can easily prove that (28.22) and (28.18) give the same results to first order in p. A CPA calculation of the reflectivity of two mixed crystals has been performed by TAYLOR (1973) on the hypothesis of mass changes only; the agreement is, therefore, only a qualitative one.

29. Mixed crystals. Mixed crystals are a certain class of disordered materials; they are mixtures AB 1 _ xCx of two AB and AC semiconductors or insulators. An important property of these materials is that the AB 1 _ xCx

crystals have the same crystal-lattice structure as the AB and AC crystals. This is to say that there is still structural order in the mixed crystal and there is disorder only with respect to the composition within one sublattice. This is essentially what has been assumed in the previous section, where structural disorder was excluded.

Usually it is understood that the basic materials AB and AC mix well over the total range of composition 9 x (0 <x < 1) of the AB1 _ xCx mixed crystal. The atoms or ions Band C seem to be distributed randomly over the sublattice, thus forming a homogeneous mixed crystal. This is concluded, first, from the fact that the lattice constant barely shows any variation within a given crystal and, second, from the linear variation of this lattice constant with concentration x. We will come back to this question briefly at the end of this section.

Much of what is said here can also be found in a review article by CHANG and MITRA (1971) to which we refer those readers who are interested in more details. A short review is also given by LUCOVSKY et al. (1968). For the quaternary compound Al_xBxC1 _ y Dy see, for example, SEN and LUCOVSKY (1975).

a) One- and two-mode behaviour. Primary interest originally has been centered on the change of the Reststrahlen frequency with composition. First investigations of the reflection spectra of these alkali-halide mixed crystal systems by KRUGER et al. (1928) showed a typical transverse optical (TO)

9 Identical with the concentration of constituent C; following common usage x is used here for what was denoted by p in Sect. 28.

C <lJ u

~ 0..

a::

Sect. 29 Mixed crystals 395

100

100

80

60

40

20

)(=0.635

x=0.50

120 160 200 240 80 120 160 200 240 80 120 160 200 240 80 120 160 200 240 Wave numbers ( cm-1 )

Fig. 29.1. Reflectivity spectra of KCl,_xBrx at 300 K. (FERTEL and PERRY, 1969)

160

-- 150 IE u - 140 Vl

~ Ll E 130 :::J c <lJ > a

Variation of transverse optical frequency with composition

3: 110~L-----~------~------L-----~------~

19 20

Mole%KCI

Fig. 29.2. Variation of KCl'_xBrx transverse optical frequency with reduced mass. 0 = 300 K, x =195K, e=80K. (FERTEL and PERRY, 1969)

Reststrahlen band for the AB1_xCx crystal over the whole range of composition. The frequency was found to shift linearly with concentration between the corresponding frequency of the AB and the AC crystals. Subsequent investigations essentially confirmed the findings of KRUGER et aI., but revealed some deviations from linearity. An example of a typical reflection spectrum and the extracted change in frequencies is given in Figs. 29.1 and 29.2. From the LyddaneSachs-Teller-(1941)-relation and from the linear variation of the high-frequency


dielectric constants Boo (CLARK and HOLONY AK, 1967; for GaP! _xAsx, for example) and Bo or, at least, of their ratio Boo/Bo (GEICK and HASSLER, 1969; for Se - Te mixed crystals), a linear variation of the longitudinal optic (LO) frequency, too, is calculated. This has been extracted also from the minimum of the reflection spectrum, since in alkali halides the LO mode cannot be observed directly in either infrared-absorption or Raman-scattering experiments. From experiments using Raman techniques, first performed by STECHANOV and ELIASHBERG (1961), the single-mode frequencies have to be extracted from the two-phonon combination bands. In crystals with the zincblende and wurtzite (or more complex) structure, the LO modes can be directly observed in Ramanscattering experiments. The rare-earth halides have a Raman-active oscillator, too, and so have their mixed crystals. As a matter of fact, the variation of the mixed-crystal Raman mode can be observed with great precision. The results are shown in Fig. 29.3. In the intermediate mixing range no remnants whatsoever of the pure-crystal Raman lines can be seen, see Fig. 29.4. From the previous

330

320

310

_ 300 "re u

260

250

240 ro ~ ~ ~ 100

x=% CONCENTRATION

Fig. 29.3. The first-order Raman frequency of CaxSrl _xF2 and SrxBa1 _ xF2 mixed crystals as a function of the concentration x. In all these crystals, there is only one first-order Raman vibration.

(CHANG et a!., 1966)


sections one would expect the linewidth to increase with increasing "impurity" concentration, which in fact is observed and shown in Fig. 29.5.

Most of the crystals with rocksalt structure exhibit the (almost) linear variation of the TO and LO mode frequency pair. Most of the zincblende- and

~Or-~Ir------TI------T-I ------~I -------r-I----~-------T----'

280 290 JOG 310 320 340 em-I

4.0-

'" 3.0-9

'" ... z a2.0 -u .-

~// 1.0- _ .-/'

. - .----o~

I lS'Fz

I I ~ ... ~

~ ... ... ~ ... ;,;

/ ___ LINE WIOTH

I I I

~ '" -:!. ... ... ... ). ( 1\)'"

-

-

, .... -

"'\.

'.. .... "" ............ --, ..... "-_ . -....-_ .. -........-

I I I CoFz I

I I I ... ~ * '" ... ...

i e '" ... ... ... ... ... .... ... ~ ~ ~ ...

Fig. 29.4. A typical Raman spectrum of the Ca0.4SrO. 6F2 crystal which was searched for addi· tional lines. Note that there is only one Raman peak, and this is between that of pure CaF2 and pure SrF2 (indicated by arrows). Data are obtained from a multichannel analyzer. (CHANG et aI.,

1966)

11.0,----.---r----,---,.-----r--"T""---,r----r--.----,

16.0

I ~.O

14.0~

:i::- 13.0 ::> 'E <I u a: -

:r 11.0 a: ... ""0

~ ~ 11 .0 o ~ ... ~ II>...J 10.0 a:

9.0 I--

•

•

• •

10 20

• •

•

• •

•

30 40 60 10 10 !O 100 X, % CONCENTRATION

Fig. 29.5. The total linewidth measured at half-intensity of the first-order Raman scattered light from the CaxSrl _ xF2 mixed crystals at room temperature. The instrumental linewidth is 4.7 cm -I .

(CHANG et aI., 1966)


100r---------------------------------~

80

~ GO

20

100 130 160

\4+--++-y = 0.75

E lie ISoK

Sect. 29

Fig. 29.6. Experimentally determined far-infrared reflectivity spectra of CdS and four mixed crystals of CdS 1 _ ySey at 15 K with Ell (c axis). A smooth line was drawn through experimental points.

(VERLEUR and BARKER, 1967)

wurtzite-type mixed-crystal systems, instead of showing one pair of frequencies, show two pairs, one for each of the mixing components, with the intensity of each pair scaling with concentration: Upon substitution of a lighter ion C for B in an AB crystal, a local mode is created. With increasing concentration the local mode intensity increases, the absorption line splits and finally turns into the AC crystal LO and TO mode lines, while the LO and TO modes of the AB crystal merge into a resonance due to B "defects" in the AC crystal. A typical reflection spectrum is shown in Fig. 29.6; the corresponding Raman spectrum is shown in Fig. 29.7, where the change in frequency and intensity is very obvious. The variation of frequency with composition is shown in Fig. 29.8.

The two different kinds of mixed-crystal behavior have been termed one-(I)mode and two-(II)-mode behavior because of the appearance of one or two pairs of frequencies, respectively.

Two-mode behavior was first observed in InPl_xAsx mixed crystals by OSWALD (1959). In fact, this behavior was observed prior to Oswald in Ge1 _ xSix mixed crystals (BRAUNSTEIN et aI., 1958). (We want to include these as mixed crystals, too, as well as mixtures of three- or more-atomic crystals.) Later, a three-mode behavior has been reported for Ge 1 _ xSix mixed crystals (FELDMANN et aI., 1966): There are three peaks in the Raman spectrum whose intensities vary with concentration and whose frequencies as a function of concentration are shown in Fig. 29.9. The three lines are assigned essentially to Si - Si, Si - Ge, and Ge - Ge pair vibrations. In fact, the intensities of the corresponding spectral lines are nearly proportional to the statistical occurrence of these pairs, (1_X)2, 2x(1-x), and x2, respectively (RENUCCI et aI., 1971). This is demonstrated in Fig. 29.10.

Returning to mixed crystals of other than the group-IV elements, there is finally a group of crystals which show one-mode behavior over part of the composition range and two-mode behavior over the other ("synthesized" mode behavior), as first reported by BRODSKY and LUCOVSKI (1968) for Ga 1 _)nxAs mixed crystals. Reflection curves and the extracted frequencies are shown in


CdS.

Fig.29.7. The Raman spectra of CdSxSe1 _ x at 80 K. The bottom trace shows the LO and TO modes of CdSe. As the S concentration is increased, these two modes converge and diminish in intensity. The top trace shows the LO and TO modes of CdS. As the Se concentration is increased,

the two modes converge and diminish in intensity. (CHANG et aI., 1969)

Figs. 29.11 and 29.12. Hence the interest has, at least for some time, been shifted to the reason why different mixed crystals show different mode behavior.

b) Theoretical models. The first theoretical study of the mixed-crystal TO frequency was made by MATOSSI (1951) who used a periodical ABACABAC ... linear chain representing an Nao.sKo.sCI crystal. Of the two infrared-active modes of this linear chain, one lies in the desired frequency region between the NaCI and KCI TO frequencies. This mode has a much higher oscillator strength than the other mode (which is at lower frequency). A more complicated linearchain model for CdS1_xSex has been used by LANGER et al. (1966) to account for the two-mode behavior. A finite mixed linear chain for both one- and two-mode behavior has been used by HASS et al. (1969).

The simple one-mode behavior of some of the AB1_xC" crystals suggests the application of the virtual-crystal model, in which each lattice site of the B

400

'E u

-..... .c (j)

200


Raman shifts of

CdSx Se1- X

100L-____ ~ ______ L_ ____ ~ ____ ~ __ ~_x

o CdSe

0.25 0.50 0.75

Sect. 29

Fig. 29.8. The frequency of the LO and TO modes of CdSxSe l -x extracted from Fig. 29.7. As the S concentration is increased, the LO and TO modes of CdSe converge to form the gap mode of

Se in CdS, while the local mode of S in CdSe splits to form the LO and TO modes in CdS. (CHANG et a!., 1969)

and C sublattice is occupied by an "average" atom with mass (l-x)mB+xmc, where mB and mc are the masses of the B or C ions or atoms (LANGER, 1961). Analogously, the coupling constant between the sublattices is given by 1/fAB1 _ x cx =(l-x)/fAB+x/f~c (POON and BIENENSTOCK, 1966a, b).

CHEN et al. (1966) proposed the so-called Random Element Isodisplacement (REI) model in which three sublattices made up of A, B, and C atoms move as rigid units (one is interested in the q=O modes) against each other. There are three equations of motion with three intersublattice force constants. One of these, the one between the Band C sublattice, is kept as a parameter that can be fixed such that good agreement with one- or two-mode crystal behavior can be obtained. (If one had restricted the Band C sublattice to moving as one rigid unit, the virtual-crystal approximation would have been recovered.) A similar model was proposed by BARKER and VERLEUR (1967). CHANG and MITRA (1968) modified this model (MREI) mainly to include the electrostatic interaction and to fix the B - C intersublattice force constant from AC and BC crystal properties (i.e. for x = 0 and 1). If allowance is made for relative motion within the three sublattices, vibrational modes other than at zero wave vector can, in principle, be calculated (CHANG and MITRA, 1971).

CHANG and MITRA (1968, 1970) used their model to support the scheme according to which one- or two-mode or "synthesized" one- and two-mode behavior could be predicted. This scheme as suggested by BRODSKY and LUCOVSKY (1968; BRODSKY et aI., 1970) is shown in Fig. 29.13. Let the mixed crystal be AB1_xCx, and let mB > mc. If a local mode of C in AB is allowed and if a "gap" mode of B in AC is allowed (see Tables 20.1 and 20.2), then the


525r----,----~--_,----_,--~

• 77 K 500

o 300 K

475 Si-Si

300

280~--~----~--~~--~--~ o 1~

Ge x Si

Fig. 29.9. Frequencies of the Ge - Ge, Ge - Si, and Si - Si peaks in Si 1 _ xGex at room and liquidnitrogen temperature as a function of silicon concentration x. (RENUCCI et aI., 1971)

8

• 6 I (Ge-Ge)

0 I (Ge-Si)

4

o 0

2

~

0 • 0 0.2 0.6 0.8 1.0

xfor Ge-Ge, I-x for Si-Si

Fig. 29.10. Ratios of intensities of the Ge-Ge peaks and the Si-Si peaks to the Ge-Si peaks in Si 1 _ xGex as a function of x and i-x, respectively. The solid curve is the theoretical function

(1-x)j2x. (RENUCCI et aI., 1971)


0.80

0.60

0.40 X=I.OO

0.20

0.0 I...-.....l.-L-L--L--L-.l..-L-.l--L---'--'---'--'-"-L-.l--'--'---'

100 120 140 160 180 200 220 240 260 280 WAVENUMBER (em-l)

Sect. 29

Fig. 29.11. Reflectances as a function of wave number for the alloy system Ga,_)nxSb. (BRODSKY

et a!., 1970)

AB1_xCX mixed crystal will show two-mode behavior. If neither local nor gap mode is allowed, then the crystal will show one-mode behavior. If either mode, but not both, is allowed it will show partly one-mode and partly two-mode behavior. CHANG and MITRA (1968) express an analogous and probably equivalent rule in terms of masses m and reduced masses 11, and predict one-mode behavior if both mB > 11 AC and mc > 11 AB' and two-mode behavior if mB > 11 AC and mC<IlAB (mC<mB)'

GENZEL et al. (1974; GENZEL and BAUHOFER, 1976) proposed another modification of the Random Element Isodisplacement Model in which local electric fields are included. They obtain four different types of mode behavior, the

0.0

Sect. 29 Mixed crystals

24D~--~--~----~--~~--~

230.

220.

-- 210. 'E

u .

>. ~ 20.0. OJ ::l <T OJ

U:: 1 go.

180.

170.

• •

0..0. 0..2 GaSb

• o TO Phonons _!lLO. Phonons

0..4 0..6

Fraction InSb

to. InSb

403

Fig.29.12. Compositional dependence of the LO and TO phonon frequencies for Ga1_)nxSb as deduced from the oscillator fit analysis. The points are derived from the data analysis. The lines

are drawn to emphasize trends in the data points. (BRODSKY et aI., 1970.)

Gap mode and local mode

both allowed

gap mode

Only local mode allowed

local mode

Only gap mode allowed

LOll)

gap mode

05 1.0 0.0 0.5 1.0 00 0.5 1.0 0.0 Composition

Fig. 29.13. General types of mixed-crystal behavior. (BRODSKY et aI., 1970.)

Neither local nor gap mode

allCMed

05


fourth being a variant of the one-mode type in which the two constituents B and C of the AB l-xCx crystal vibrate against one another in an acoustical mode and cause a low-intensity band, as is found in NH4Cl1 _ xBrx mixed crystals (BAUHOFER et aI., 1974a, b, 1976; PERRY et aI., 1977). The distinction then is in optical and acoustical one- and two-mode behavior in various parts of the Brillouin zone. Information of the end-member crystals AB and AC only are required, and the model calculations compare well with experimental data. Using this model, MARTIN (1975b) has calculated oscillator strengths in fair agreement with experiment. FUJII et al. (1983) employ the resonant Raman scattering to study zone-boundary modes in mixed silver halides.

To obtain more quantitative results, VERLEUR and BARKER (1966) used a complicated "cluster" model, in which short-range clustering and long-range randomness are generated by introduction of an order parameter which weights the force constants between nearest and second-nearest neighbors.

Early low-concentration Green-function treatments (ELLIOTT and TAYLOR, 1967; JASWAL and HARDY, 1968a, b) have been able to give only qualitative agreement with experimental results for LiH 1 _ xDx (MONTGOMERY and HARDY, 1965), while good agreement for the frequencies and integrated absorption intensity was obtained for the GaAsO.94PO.06 system (ELLIOT and TAYLOR, 1967).

PERSHAN and LACINA (1968, 1969) and LACINA and PERSHAN (1970), using the random-impurity formalism in the low concentration limit, were able to account for about 60 % of the concentration-dependent line broadening in the Ca1 _ xSrxFz system, as was shown in Fig. 29.5. The calculation reproduces the one-mode behavior and, in fact, seems to give rather a good account of the concentration-dependent shift. PERSHAN and LACINA point out in this connection that the phonon self-energy (f, see Sect. 28b, due to defects is related to the 1- or II-mode behavior of the mixed crystal. To lowest order, (f is proportional to the scattering matrix t, (28.6'),

(f(1)=xt=xv(1 + go V)-l.

If go v is "less" than 1 (in the sense that det (1 + g~ v) ~1), then

In this case the self-energy causes a shift of the phonon self-energy that is linear in the concentration; the lowest-order contribution to the imaginary part would be

Im(f(l)~ -xvg~v

which is "much less" than v. On the other hand, if there is a frequency at which det (1 + g~ v) equals zero,

then a resonance is created and a new mode with concentration-dependent intensity appears: the crystal shows two-mode behavior. Of course, in their simple form the arguments above are valid only at low concentrations and are able to predict only the initial (low-concentration) behavior. VINOGRADOV (1969) has used the low-concentration Green-function theory to predict localand gap-mode frequencies and thus one-, two-, and synthesized-mode behavior


along the lines of argumentation of BRODSKY and LUCOVSK Y (1968). TAYLOR (1973) has performed a CPA calculation of the reflectivity spectra of KBr1_xClx (one-mode) and of K1_xRb) (two-mode), which reproduces the qualitative features of the experimental data, but lacks quantitative agreement, probably because of the neglect of force-constant changes, as TAYLOR points out.

c) Changes in the lattice constant and 1 vey relation. Pressure experiments have been performed with mixed crystals of different mode behavior, i.e. KCl1_xBrx' ZnS1_xSex, CdS1_xSex' and K1_xRb) (FERRARO et aI., 1970a, b, 1971). The interesting result is that the change in e.g. the TO frequency seems to be due essentially to the change in the lattice constant, irrespective of whether this was achieved by applying hydrostatic pressure or by varying the relative concentration of the constituents. This indicates that the effects of local impurity-induced distortions of the lattice, given by the term gCPl in (31.33) below, are small. The experimental results are shown in Fig. 29.14.

Knowing that the change in frequency is proportional to the lattice constant (Ivey relation), the contribution of the thermal expansion to the anharmonic shift of the U -center localized mode can be deduced from the variation of the Ucenter local mode with lattice constant, which in turn can be varied either by applying pressure, or by changing the relative concentration of constituents of a mixed crystal in which the U -center is embedded. U -center local modes in additively doped crystals have been observed by SCHAEFER (1960); MIRLIN and RESHINA (1966); BARTH and FRITZ (1967); YATSIF et aL (1967); JONES et aL (1968, 1969); ASHBURNER et aL (1968); CHAMBERS and NEWMAN (1969, 1971a, b); CHAMBERS (1971); CASTRO et aL (1975); and KOSTER and DEN HARTOG (1977); a theoretical investigation of these defect pairs has been carried out by JASWAL (1966); STRIEFFLER and JASWAL (1969b); SCHNEIDER (1973); GUPTA (1974); and KRISHNAMURTHY and HARIDASAN (1978).

2.22 o

2.18

en .9 2.14 •

2.10

•

• 0.001 kbar x 2.77 k bar o 831 kbar j\, 11.08 kba.r o 16.62 kbar

-+- Est.error

o • x

• x

2.06'-:-::-___ ~-----:o-!c:_---__=_'=,______--~ 0.48

Fig. 29.14. Plot of VTO vs. lagro for KCl1_xBrx at various pressures. (FERRARO et a!., 1970a)


The Li + ion in potassium halides is known to produce a low-frequency resonance. The Li + ion is at a defect site in KBr while it has an off-center position in KCI and KI. CLAYMAN et al. (1967, CLAYMAN and SIEVERS, 1968 a) have tried to make the Li + equilibrium position in KBr unstable by alloying KCI or KI to KBr and thereby changing the KBr lattice constant. They did observe a shift (and strong broadening) of the KBr: Li + resonance; however, it turned out that with the mixing ratio (x ;:;::;0.1) at which the instability is expected to occur the conventionally grown KBr1_)x crystal was not homogeneous. X-ray scattering experiments on KBrxIl -x by NAIR and WALKER (1971, 1973) revealed one lattice parameter, changing linearly with small concentrations x. However, in the regime 0.3 < x < 0.7 three lattice parameters were observed.

30. Anharmonic effects in perturbed crystals

a) Introduction: Resonance modes in analogy to the Reststrahlen or Raman oscillator. Anharmonicity of the crystal potential causes two important phenomena. The interaction of the phonons produces effects connected with temperature, like thermal expansion of the crystal and lifetime and energy shifts of the phonons. Related effects are observed if, instead of changing the temperature, external fields are applied which change the relative distance between the ions. These effects show up in the perturbed crystal as they do in the pure crystal. They are the subject of this and the following section.

Primary interest is in resonance (localized or resonant) modes. These exhibit many similarities with the Reststrahlen (or Raman) oscillator and therefore will be discussed in close analogy with the latter. A resonance mode shows in an infrared-absorption of Raman spectrum a predominant peak whose position, width, and intensity vary with temperature.

The main difference between the Reststrahlen (or Raman) oscillator and a resonance mode is that the former is dominant because of very strict selection rules in the pure crystal while the latter shows a strong peak due to localization of vibrational energy. The details will be worked out in the following Sect. 30b.

The temperature-dependent properties of a resonance mode can be interpreted in terms of the self-energy. Limiting oneself to high-frequency localized and low-frequency resonant vibrations one knows that in these cases the eigenvectors are extremely large in a small region about the defect. This facilitates the discussion and agrees with the body of experimental and theoretical investigations which are restricted essentially to resonance modes due to defects in alkali halides.

Like for the Reststrahlen oscillator, the self-energy of a resonance evaluated at the resonance frequency will give the resonance-line intensity, width, and shift as a function of temperature, while the multi-phonon (sideband) spectrum will be obtained from the self-energy evaluated at frequencies other than the resonance frequency. .

A number of approximations for the self-energy will be necessary to enable one to discuss the temperature-dependent properties of perturbed phonons and, in particular, of resonance modes. These are: 1) the reduction of the self-energy matrix to a finite size which allows qualitative considerations; 2) the selection of

Sect. 30 Anharmonic effects in perturbed crystals 407

processes contributing to the self-energy; and 3) the truncation of a Taylor-series expansion of the self-energy for frequencies close to the resonance frequency. These approximations will be the subject of Sects. 30c, d, and e. Some attention will also be given to the simultaneous effects of concentration and the anharmonic and perturbed anharmonic self-energy. The results will be very approximative, but will, together with those of Sect. 31 for the thermal expansion, shed some light on the quasi-harmonic approximation for perturbed crystals (Sect. 30c). Intensity, shift and width of a resonance-mode line will then be considered in Sects. 30f, h, and i, with a sidestep to isotope effects in Sect. 30g.

Up to this point the self-energy at or very close to the resonance frequency has been considered. At more distant frequencies, the self-energy gives rise to combination bands which will be discussed in Sect. 30j. Higher-order effects, finally, will be considered in Sect. 30k.

In this section no consideration will be given of the shell-model aspects of the anharmonic theory, since the structure of these results is given in Sects. 27 e and f.

The similarity between external field-induced strain and that due to thermal expansion, and the influence of both on phonon frequency shifts, suggests a combined treatment. Strain-induced phonon-frequency shifts deserve special attention and are formally sufficiently different from the self-energy shifts to justify a separate treatment in Sect. 3l.

It will be impossible to give a complete account of experimental and theoretical investigations of anharmonic effects. Selected contributions will be cited as appropriate. Theoretical work has mainly followed experimental work. Experiments have for a long time focused on resonances rather than on overall changes in, say, a two-phonon spectrum caused by breakdown of the quasimomentum conservation law. Even first-order spectra, which are forbidden in cubic crystals (except, possibly for the Reststrahlen or Raman oscillator), have been obscured by many-phonon spectra and background noise unless some predominant absorption lines appeared, usually caused by a resonance (localized or resonant) mode. Also, sufficiently high impurity concentrations were necessary, as well as sufficiently low temperatures, good spectral resolution, etc.

Because of the spatial localization of vibrational displacements in a resonance mode, coupling to other modes occurs locally, especially in the case of coupling of resonance modes to resonance modes. This is of great practical importance since it reduces the number of anharmonic parameters involved to a few around the impurity. As a matter of fact, numerical calculations of the anharmonic self-energy have sometimes been able to account for experimental data by using only a single anharmonic parameter. In general, coupling of a resonant mode to a band phonon, for example, is of long range, because both the Coulomb anharmonicity and the band-mode displacements extend far into the crystal. Inclusion of the Coulomb anharmonicity in an actual numerical calculations has to our knowledge been done only once, and in that case for a pure crystal (KNOHL, 1972), the conclusion being that it changes the absolute value of the damping function but does not change its overall shape. For defects, this problem offers itself to future study.


b) Qualitative aspects of the anharmonic self-energy in perturbed crystals. As mentioned above, resonance modes display the same anharmonic properties as a Reststrahlen oscillator. In spite of many similarities, there are certain differences with three kinds of origins which, of course, are all related to the presence of impurities. In short, the self-energy of a perturbed phonon is different from that of the corresponding unperturbed one due to: 1) the reduction of the space symmetry to point symmetry, 2) the change of anharmonic coupling constants, and 3) the alteration of the harmonic lattice dynamics due to the defects.

First of all, the periodicity of the pure crystal is reduced to the point symmetry of the defective lattice. Consequently, the quasi-momentum of the pure lattice ceases to be a good quantum number. The anharmonic terms in the potential couple modes of different irreducible representations, the product of which must contain the fully symmetric representation; for pure crystals this restriction (selection rule) is on the irreducible representations of the space group (wave vector conservation), and of the point group for crystals with a defect. The selection rules are often of great practical importance for the calculation of anharmonic effects, as they are for the harmonic ones. For example, in crystals with inversion symmetry, the even or odd "parity" of a phonon displacement (Sect. 22d) is a good quantum number. The quasi-momentum conservation of the pure crystal is then replaced by the following, very simple selection rule: The product of the phonon parities which can couple by anharmonicity has to be positive.

In pure crystals with sufficiently high symmetry the self-energy for the wavenumber vector q = 0 is diagonal in the branch indices; this permits one to define the self-energy of the Reststrahlen oscillator in a sensible way. However, even though it is always diagonal in q due to periodicity, the self-energy of a purecrystal phonon in general is not diagonal in the branch indices (MARADUDIN and IPATOVA, 1968). This leads to what has been called "intermixing of normal modes". The reduction of symmetry to point symmetry in perturbed crystals causes the self-energy to "intermix" all those normal modes which transform according to the same irreducible representation of the point group.

From the preceding arguments, it is clear that in general one now can no longer talk about, e.g., "the width of a resonance" since the resonance mode becomes intermixed with all the other modes of the same irreducible representation. However, the Dyson equation for the anharmonic Green function, GA,

(30.1)

(cf. (36.75)) in terms of the harmonic Green function, GH, and the self-energy, II, shows that, whenever the frequency is at or very close to a resonance frequency, only those terms of GH will be important which refer to the resonance vibration. (The appearance of intermediate Reststrahlen or Raman oscillators makes a slightly more detailed analysis necessary. This will be done in Sect. 30c.) Because of this property of the harmonic Green function, among all the elements of the self-energy matrix only a few are relf<vant for the anharmonic Green function at or near the resonance frequency, see Sect. 30c for details.

The breakdown of the wave-vector selection rule not only changes the symmetry of the self-energy, it also causes the self-energy of normal modes to be


numerically different, since the self-energy depends on many-particle Green functions. Since the combination of the phonons involved in the Green functions is no longer restricted by wave-vector selection rules, the number of phonons which can couple through anharmonic interactions is increased. A defect which merely reduces the periodicity of a crystal to point symmetry but otherwise leaves it essentially unperturbed, should in principle lead to a change in the selfenergy, that is to say, it should cause defect-induced broadening and shift of, say, the Reststrahlen oscillator, as well as defect-induced enhancement of phonon spectra of various orders.

The many-particle Green functions mentioned above depend in real-space representation on the eigenvectors of the harmonic lattice. In the momentum representation (we will keep this name in analogy to the case of perfect crystals) the coupling matrix elements rather than the Green function depend on the eigenvectors. In any case, the presence of defects in the crystal does change eigenvectors which thus secondarily change the self-energy, over and above the change due to the breakdown of the quasi-momentum conservation.

Finally, the coupling constants for the interactions between the phonons will be changed locally. (The real-space representation is adopted here, so the change in eigenvectors will alter the Green functions only.) This is due to a different potential between the defect and the lattice, as described in Sect. 22a. In addition, the distortion of the crystal due to the presence of the defect will lead to anharmonic potential expansion coefficients which are different from those of the host crystal, following arguments similar to those of Sect. 22 b.

c) Diagonal and off-diagonal elements of the perturbed self-energy. The anharmonic width and shift of a phonon and the temperature dependence thereof can be observed only by an experiment in which the phonon under consideration is excited or destroyed by an external probe. In Sect. 25 the cross-section for infrared absorption or Raman scattering by one-phonon processes (the latter at T=O K) was obtained, and in Sect. 28 the results were generalized to lowdefect concentrations; the cross sections were found to be proportional to a quantity S,

s=yTGY

= jiT ~0(0)Y + pLl yT g LI y (30.2) with

ji= yo+p(l-tgo) Lly (30.3) and

~ 0(0) = [(~ 0(0))-1 + pt]-1 (30.4)

where y represents either a charge or the first-order polarizability tensor. The equation can be expressed alternatively in terms of the eigenvalues and

eigenvectors of go v similar to (25.23)

with

s=LYj[{~0(0)-1}ii'+P Ibii, ]-1 Yk+PLLlYj-l/lj LlYj (30.2') jk +/li jk j +/lj

(30.3')


and (30.5)

The advantage of this representation is that the term of interest, (1 + 11)- \ is a scalar rather than a matrix. For resonances, the sums in (30.2') have their main contribution from the single term which refers to the resonance vibration (except for the term to zeroth order in the concentration).

In Sects. 25 and 28, G represented the harmonic one-phonon Green function, GH. If anharmonicity is taken into account, G is the anharmonic Green function, GA. In Sect. 34, a Dyson equation for GA is derived and solved. GA can formally be written in the form

(30.6)

In a perturbed crystal, GH as well as the self-energy II are perturbed, and (30.6) can be written in real-space representation as

cf. (23.1).

(GA)-1=G0 1+ V+ll

=G0 1+ V+llo+iJ.ll

= (G~)-l + (V + iJ.ll), (30.7)

A low-concentration analysis along the lines of Sect. 28 would treat V + iJ.ll on the same footing and identify G~ as the unperturbed Green function. The results for the one-phonon spectra then have the same structure as (30.2) and (30.3) and are obtained from these, if one performs the following substitutions,

Go=G~-.G~=(Go 1 + llO)-l,

V-. V+iJ.ll,

one then obtains in the low-concentration approximation

go = g~-. g~= [(1 + Go llO)-l Go]",

g= gH-. gA= [(g~)-l + v+ iJ.1t]-1

with ~ 0(0) -. ~~(O) = [(~~(0))-1 + ptA]-l

tA =(v + iJ.1t) [1 + g~(v+ iJ.1t)] -1,

~~(O) = [(~ 0(0))-1 +1to(O)] -1.

(30.8)

(30.9)

(30.10)

Here 1to(O) is the Fourier transform of llo for q =0 and is defined analogously to ~0(0), (25.9). For later convenience, one can define an effective self-energy matrix (in the impurity space) by

(30.11 )

similar to the definition of the effective mass cf. (24.4) and (24.7). With this in principle well-defined quantity (30.8) and (30.9) can be written as

g~=(I+go1t6)-1 go, go=g~

gA=[I+go1t6+go(V+iJ.1t)]-1 go

= [1 + go(v +1t*)] -1 go'

(30.12)

(30.13)


The main conclusion from (30.9) and (30.10) is that the harmonic and anharmonic widths of a resonance line simply superpose and that the width of the Reststrahlen or Raman mode is that of the pure crystal plus a correction, ptA, which is proportional to the defect concentration and which is a somewhat complicated function of harmonic and anharmonic perturbations. The temperature-dependent properties of the Reststrahlen oscillator are thus determined by the bulk (harmonic plus anharmonic) self-energy.

As far as the perturbed vibrations are concerned, (30.9) shows that the perturbed anharmonic Green function, GA , is given in terms of the unperturbed anharmonic Green function, G~. The latter' involves the self-energy of the pure lattice which may be of arbitrary range even if only the impurity-space components of G~ are needed. This is expressed by n6, which is different from no.

In the eigenvalue-type treatment, the substitution H A

flj=flj -'>flj

in (30.2') and (30.3') has to be performed, where the flj are the solutions of the eigenvalue equation

(30.14)

The main difference between (30.14) and the corresponding equation (23.19) for the harmonic case is that, in the latter, the different rows of an irreducible representation of the point group do not intermix, while here they do. This may be important for numerical computations. It may be desirable to treat flA as a matrix with the rank equal to the number of rows. Minor attention will be given to this point in the following.

It was mentioned in connection with (30.1) that the self-energy in perturbed crystals "intermixes" all modes which belong to the same irreducible representation, but that the Green function has appreciable contributions only at or near the resonance frequency. The exact treatment, which would require the diagonalization of the total "intermixed" system, can then be approximately solved by diagonalization of a small subsystem consisting of the "most important" terms of nand GH•

In real-space representation (30.6) reads

(30.15)

and the important terms are those elements which refer to the defect and a few of its neighbors. The number of neighbors involved is the smaller the more localized the vibration is. If, in fact, GH is limited to a small ("impurity") space, then the arguments for the localization of the T matrix, (23.5), can be reversed, and (30.15) can be solved to give

gA=[l+gHn]-l gH

=[l+(l+gov)-l gon]-l(l+gov)-l go

=[l+go(v+n)]-l go, (30.16)

as an approximation to (30.13). One may argue that gA has a priori non-zero elements almost everywhere. On the other hand, with increasing distance from


the defect, GH becomes increasingly small. In the same sense the finite range of the perturbation V was defined in Sect. 22b. Effects of non-localization of the Green function and self-energy are immersed in 1t* = 1t6 + ,,11t, (30.11), by which quantity (30.16) and (30.13) differ. For later use, the denominator common to the expressions for gA, tA, and ji can be written as

[1 + g~(V+,,11t)]-l = [1 +(1 + go1t6)-1 go(V+,,11t)]-l

= [1 + go(v +1t*)] -1 (1 + go 1t6)'

In momentum space the elements of (30.6) are

[(GA)-lJu' = [(GH)-lJu' + IIu ',

(30.17)

(30.18)

and the most important elements of G and [J are expected to be those with frequencies close to the resonance frequency. A light defect may have, in general, a (possibly degenerate) high-frequency localized mode. This is the most important contribution for W ~ W L if the local-mode frequency, W L ' is sufficiently large compared with the band frequencies. The important contributions for a resonant mode come from all those band vibrations which have a frequency close to the resonant-mode frequency. Since this number is large, the evaluation of (30.18) for resonant modes seems to be a hopeless task; the real-space representation, (30.15), is necessary, while for the high-frequency localized modes any representation may be appropriate.

Another difficult point for a practical application of (30.9) and (30.10) is the reduction of ,,1[J to a sufficiently small impurity space. The change of the selfenergy, ,,1[J, has two contributions. One comes from the change of the anharmonic coupling constants. These will not extend further into the crystal than do the harmonic force-constant perturbations, which generally determine the size of the impurity space. But even though the anharmonicity perturbation may cause, for example, the local decay of a phonon into two others, the two latter generally extend far into the crystal and, due to the unperturbed anharmonicity, may combine at any other lattice site to give the original phonon again. On the other hand, it is here that the considerations connected with (30.1) come into play. The decay and re-formation of a resonance phonon to first order is localized at one defect, since the reformation of the original resonance phonon at a different defect is of second order in the concentration.

In summary, one can say that the expansion of the self-energy in terms of the concentration is probably an extremely poor approach to the problem. Some improvements may be expected from the analysis of the vertex parts of the selfenergy rather than the self-energy itself, but even then the problem seems likely to remain rather complex.

The contribution of anharmonic perturbations to the quasi-harmonic corrections of the force constants is related to the thermal expansion, which will be discussed in Sect. 31 to a somewhat more satisfying degree of approximation than in this section. The pseudo-harmonic corrections also have contributions from the other terms of the self-energy. These remain essentially undetermined.

The subject here will be resonances in infrared and Raman spectra; the harmonic properties of resonance modes are mainly determined by just gH, as


reasoned in Sect. 25b. This is still true for anharmonic defects in the group-IV semiconductors, where the first term of (30.2) or (30.2') is identically zero. In other crystals, this first term contains the anharmonicity of the Raman or Reststrahlen mode. Generally it causes, among others, multi-phonon spectra superposed on the spectrum, as obtained from the second term of (30.2) or (30.2'). The aim here will be not so much these as resonances. The anharmonic properties of the latter are mainly a function of expression (30.17) rather than gA, and local as well as bulk anharmonicity plays a role.

d) Low-order contributions to the self-energy. A discussion of the exact selfenergy is possible only in rather general terms, not only because of the occurrence of diagonal and off-diagonal terms but because of the large number of factors which contribute to it. Taking into account the effects of only the thirdand fourth-order potential expansion coefficients automatically restricts the number of physical processes. One has further to limit the discussion to qualitative aspects of essentially those processes which correspond to results of first- and second-order perturbation theory. These processes were explained in Sect. 2b where a graphical representation was also given. The diagrams of Fig. 30.1 represent some of the low-order contributions to the self-energy: A and B denote processes which are independent of frequency and contribute to the anharmonic shift only, while C and D, which represent decay of and scattering at a particular phonon, contribute to both the real and imaginary parts of the self-energy and thus to shift as well as to width.

The discussion of the effect of thermal expansion is deferred to Sect. 31, although some qualitative remarks are found in Sect. 30g.

It is rather questionable from the standpoint of perturbation theory whether it is consistent to take into account on the one side both cubic, V3 , and quartic anharmonicity, V4 , to second order, and on the other side not to include V3 in

A B -0-

c e

o

E F G

Fig.30.1. Graphical representation of some low-order contributions to the anharmonic phonon self-energy. A represents the simplest of all those processes which correspond to first-order perturbation theory. Processes B to D correspond to second-order perturbation theory. Diagram B describes thermal strains in the crystal; diagrams C and D describe two- and three-phonon decay and scattering processes. Diagrams E to G denote higher-order processes; F is an interference

process between D and E


fourth order. In doing so, terms as for example represented by diagram E of Fig. 30.1 have been omitted. In that same order of perturbation theory, processes as represented by diagrams D and E interfere, the interference being represented by diagram F of Fig. 30.1. There are a number of additional processes which are a certain iteration of diagrams A, B, and C (just as E is an iteration of C in C). In fact, higher-order approximations seem to be important, one will come back to that point later in connection with the U -center sidebands and the resonance mode of the system NaCl: Cu + (Sect. 30k). In the following let the different processes represented by diagrams A, B, etc. for simplicity referred to as processes A, B, etc.

While investigating the width and shift of a local mode, i.e., while considering the diagonal and possibly off-diagonal elements of the self-energy matrix of a localized mode, the representation in momentum space may be more useful than in real space. In this representation the self-energy is

lIA*A,=~(hj2wA)1/2 L x*(L,IXIA) fI LaL' a'

. II(L, IX; £, IX') X(£, IX' I A')(hj2wA,)1/2. (30.19)

The self-energy is given in this form in Sect. 38a. If A and A' refer to an Einstein oscillator (which may be degenerate), one finds, incidentally,

2 WE II EE' = II (0, IX; 0, IX')! M (0).

The diagrams A to D of Fig. 30.1 represent a selection of processes contributing to the self energy. These processes have been considered in Sect. 12e. Even if they do not represent a systematic collection from the view point of perturbation theory they are, however, the qualitatively most different of the loworder terms. According to Sects. 12e and 38 a the contributions of processes A to D (Fig. 30.1) to the self energy are

IIA*A' = LI A* A' - irA* A'

=1 L {V4 (A*, A', Ai' An- L V3 (A*, A', A2 ) ~ V3 (Ai, Ai' An} (1 +2n i ) Al A2 W 2

-1 L V3 (A*, Ai' A2 ) g2(Ai , A2 ) V3 (A', Ai, Ai) Al A2

--1; L V4 (A*, Ai' A2 , A3 ) g3(Ai , A2 , A3 ) V4 (A', Ai, Ai, A~) (30.20) Al A2 A3

with

(30.21)

(30.22)


Here v.. are the anharmonic potential expansion coefficients and g2 and g3 are retarded two- and three-phonon Green functions in the harmonic approximation. In real-space representation one finds from (30.19)

n(l,1')=h I,{<1>4(1,1',1", I"') l"l'"

- I, <1>3(1,1', m) GST(m, m') <1>3 (m', I", I''')}!(v(l'') v(l"') mm'

--2h L <1>3(1, m, n) G2 (m, n, m', n') <1>3 (m', n', 1') mm'nn'

h2

3 ' I, <1>4(1, m, n, p) G3(m, n, p, m', n', p') <1>4(m', n', p', 1') . mm'nn'pp'

(30.20') where

!(v(l) v (I') =! I, x(ll A) wi 2 X*(I' I A) hw;.(!+ nJ) ;.

=~ S dw 1 +n(w) G"(I, l'1 w) n -00 w

=-21 S d~2 G"(I, l'1 w) E(w) now .

(30.24)

with

G" is the imaginary part of the one-phonon (Lifshitz) Green function, and G2

and G 3 are two- and three-phonon Green functions in the Hartree approximation, which can be written as (cf. (38.18))

1 1 1 OOs 1 OOs (1 +n l ) ... (1 +nn)-ni ... nn Gn(I"'" n,ml,···,mn)=- dw l ···- dWn .

n -00 n -00 WI + ... +Wn-W-IB

(30.22')

The self-energy was given by MARADUDIN (1963) in the form of (30.20). Prior to this KLEMENS (1961); MOZER and VINEYARD (1961); MOZER (1963); KRIVOGLAZ (1961a, b); and KAGAN and IOSILEVSKII (1963 a, b) considered the contributions to the width of a local mode from process C alone. Since twophonon decay processes could not account for the width of the high-frequency localized mode of the Ucenter because of energy conservation, three-phonon decay processes (D) seemed necessary (MIRLIN and RESHINA, 1964a) and were


added by IPATOVA and KLOCHIKHIN (1966). Five-phonon processes were considered by ZAVT et al. (1965); see also WILSON et al. (1968). Process B, which contributes to the shift to the same order of perturbation theory as C was also considered by KRIVOGLAZ (1961 a, b, 1964).

None of these papers gave any attention to the appearance of the resonance oscillator in the final state of decay which may occur if the resonance phonon does not decay into three phonons but the resonance phonon and, say, a band phonon combine to give the resonance phonon again and another band phonon. This process is described by the difference (or scattering) part of process D. In other words, it describes scattering of band (or even resonance) phonons at the resonance phonon. The importance of this term for the line width of the Ucenter local mode was first pointed out by HANAMURA and INUI (1963) and ELLIOTT et al. (1965). It was subsequently investigated by IVANOV et al. (1965, 1966) and BILZ et al. (1966). A quantitative calculation of the line width of the U-center local mode was given by GETHINS (1970). Resonant modes have been treated numerically by TIMMESFELD and BILZ (1968); TIMMESFELD (1968); and BENEDEK (1968). The change of the Tl + -induced Raman spectra with temperature (and pressure) was investigated by TAYLOR et al. (1975); they find that the volume dependence account for most of the changes and that, therefore, the contributions from the genuine multi-phonon effects cancel; also, the anharmonic forces are essentially those of the pure crystal in accordance with the fact that the Tl + defect in alkali halides seems to represent an isotopic defect.

e) Approximate form of the anharmonic Green function. In this section the Green function will be approximated by terms which will be valid either very close to or very remote from a resonance frequency. In this and the following sections an Einstein-oscillator resonance will often serve as an introductory example, and then generalizations to extended resonances will be made. In accordance with the considerations in Sect. 23c the anharmonic resonance frequency, WR , is defined by the vanishing of the determinant of the real part of the denominator of the anharmonic Green function,

(30.25)

Mixing of a high-frequency localized mode with other modes can possibly be neglected, whence (30.25) in momentum space representation, see (30.19), reduces to

(30.26)

For an Einstein-oscillator mode one uses the localized nature of the Green function (rather than the "perturbation", A) in real space and finds from (30.25) and (30.13)

[g(O, a; 0, a)] -1 + 6(0, a; 0, a) =(w~ - w~) m + 6 =0, (30.27a)

which is anologous to (30.26). Somewhat more generally, the resonances of gA are given from (30.13) by

0::::; Re(1 + go(v+ n))= 1 + g~(v+6)+ g~ y (30.27b)

Sect. 30 Anharmonic effects in perturbed crystals

with go from (24.14). For localized modes, (30.27b) reduces to

1+g~(v+(j)=0

417

(30.27 c)

(IPATOV A and KLOCHIKHIN, 1966) which is, of course, identical with (30.27 a) for an Einstein oscillator. For frequencies close to wR one finds an approximation for the Green function (30.18) if one neglects the off-diagonal elements of n H' (for the inclusion see (38.32) ff.) and takes into account the frequency dependence of the shift function LlRR by a truncated Taylor expansion .

LlRR(W) ~ LlRR (wR) + (WZ - w~) aLlRR(W)/awzl ro = WR

=LlRR(WR)+(WZ-w~)J~, (30.28)

whence, to first order, G~R is given by

GA ~ 1 _ 2WR_ RR 1-2wRLl~ wi-wz-i2wRrR

(30.29)

with (30.30)

Compared with a zero-order approximation in which the frequency dependence is neglected, J~ = 0, the area under the imaginary part of G~R' as given by (30.29), as well as its width is reduced by a factor of

1 (30.31)

1-2wRJ~ .

Kramers-Kronig properties of the self-energy show that

_ a I Ll~=awz (LlRR(W)-LlRR(OO)) ro=WR

=_13_ ~ p <XlJ rRR(Q) dQzl = _~ p <XlJ IitR(Q) dQz awz'It 0 wZ_Qz ro=WR 'It 0 (wi-QZ)Z

is negative. (Figure 30.2a shows an example of an experimentally determined shift function of a localized mode.) The factor (30.31) leads, in fact, to a reduction. The upper limit of integration here is a frequency that is very large compared with any phonon frequency but small compared with an electronic excitation energy.

For the validity of truncating the Taylor expansion for LlRR(W), (30.28), one has to have sufficiently small higher-order derivatives. From general considerations about the properties of the Kramers-Kronig transform, it follows that this occurs if rRR(W) has a minimum close to or at the resonance frequency WR or wR. This also allows one to neglect the frequency dependence of rRR(W) in the region of the resonance frequency, as was done in (30.29).

Starting from (30.29) we see that the intensity, width, and shift of the resonance are given approximately by the expressions (30.31), (30.30), and (30.26), or (30.39) below.

Giving up the rather artificial concept of an Einstein oscillator, which has been pursued so far, one finds the anharmonic resonance frequency in general


from (30.32)

for a particular j= R. Equation (30.32) is equivalent to the real-space form of (30.25) and is the anharmonic generalization of (24.10). From (30.32) and (30.12) for the linear case (30.27 b) is recovered.

Expanding /l~(w) around w2 = w~ as above and as in (24.12), one obtains the Green function approximated by a Lorentz curve for w 2 ;:;::; w~,

/lR 1 1 1 l+/lR ;:;::;-l+/lR ;:;::;a/l~/aw2 w~-w2-i2wRYR

(30.33)

with 2WR YR = /l~/(a /l~/aw2) (30.34)

and /l~ and a /l~/aw2 evaluated at w2 = w~. It should be noted that the particularly simple form of (30.2') together with (30.14) is directly accessible to numerical computation, but is practical only if the "defect space" is small. The approximate form will, however, be useful for qualitative discussions of resonance vibrations.

On the other hand, for frequencies sufficiently remote from the resonance frequency WR , g-l becomes sufficiently large compared with n, so that (30.9) may be expanded in terms of n into some low order,

gA;:;::; gH_ gHngH

which agrees with results of low-order perturbation theory for one- and multiphonon projected densities. The second term is responsible, for example, for the absorption due to multi-phonon processes (sidebands, overtones, etc.). The corresponding expansion in the momentum representation becomes rather questionable if only the resonance-mode term of the self-energy is used, because Einstein-oscillator properties may no longer persist.

Example: The U center in alkali halides Figure 30.2 shows the damping and shift functions of the system KCI: H -,

calculated from the experimental absorption constant (dashed curve). The calculation is actually approximate; if higher-order corrections are included (ZEYHER and BILZ, 1969) the damping and shift functions become scaled by a factor of about 2. The qualitative behavior is, however, unaltered.

The shift has a negative slope near the resonance frequency which has the consequence that the factor (30.31) is indeed less than unity. Also, the damping function has little curvature near the resonance frequency; this indicates that the truncation of the Taylor expansion is a good approximation.

The damping function is, in fact, relatively small near the resonance frequency. The calculations are not sufficiently accurate to allow conclusions to be drawn as to whether or not it is a good approximation to neglect the frequency dependence of the damping function at the resonance frequency. DOTSCH

(1969 a) finds the same kind of behavior in the system NaF: H -. f) Intensity of resonances. The area under the curve which is described by

the imaginary part of the anharmonic Green function, (30.29), is

Ws 1 2WR YR d 2 _ 2WR n _ o 1-2wRj~ (W~_W2)2+(2wRYR)2 W -1-2wRj~ (1 + o (YR/WR)).


Wave number

350 400 ~50 500 550 600 650 cm-1 700

I , I

-- K(vl, 17K I - v·r(v) --4-----+1, H------jl--ft-----ji---i-----i ,

I I

Local mode peak § x100 I ti I

a

~ 5~----+_----4-----+'»4----1r~~~~.__r----~ (Jl c

'0.. E o o

o 350

~ 5 III

>- 0 u c ~ -5 CT <II

tL -10

550 600

-15~--~--~---~----~----~----~--~ 350 400 450 500 550 600 650 cm-1 700

Wave number

419

Fig. 30.2 a, b. Damping and shift function of the KCl: H- U-center local mode; (a) dashed line, absorption constant; full line, damping function; (b) shift function. (BILZ et aI., 1966)

The correction of the order of the relative width is usually small for the weakly anharmonic insulators and semiconductors. It has been shown above that the factor (30.31) leads to a decrease in the intensity compared with the harmonic intensity. As the temperature is raised 1.a~1 increases, thus further reducing the intensity. .

On the other hand it can be shown generally (see, for example, HUGHES,

1968; STRAUCH, 1969; ALEXANDER et al., 1970) that the total integrated area is constant,

1 ooJ A 2 - ImGRR(w)dw =2wR • 1l: 0 .

(30.35)

This is the momentum-space analog of what is usedin (25.7'). The integration is performed over both positive and negative frequencies corresponding to phonon absorption and emission processes, and either giving the same contribution. The integrated absorption intensity then depends only upon the charges and masses of the ions and not upon temperature. The intensity under the main resonance line in an infrared or Raman spectrum which decreases with increasing


temperature has, therefore, to appear somewhere else in the spectrum. In fact, it will be shown in Sect. 30j (see also below in this section) that the main-line intensity decreases to the same extent to which the many-phonon (sideband) spectrum increases (INUI and HANAMURA, 1962; ELLIOTT et aI., 1965; HUGHES, 1968; STRAUCH, 1969; IPATOVA et aI., 1968, 1969 a). Processes A and B which contribute only to the shift are frequency-independent. They contribute neither to the change of intensity nor to the multi-phonon spectra.

If 2WRJ~~1, the factor (30.31) may be approximated by a kind of DebyeWaller (or Huang-Rhys) factor,

(30.36)

As the "optical analog of the Mossbauer effect", the exponential has attracted much interest (MCCUMBER, 1964a, b; LUBCHENKO and PEVLIK, 1964; TAKENO and SIEVERS, 1965; MITRA and SINGH, 1966; HUGHES, 1968; KLOCHIKHIN and SUBASHIEV, 1968; IPATOVA et aI., 1968, 1969 a, b; SILSBEE, 1969; STRAUCH, 1969) and has been the source of much controversy as well. In the form of (30.31), this factor was obtained by IPATOVA et al. (1968, 1969a) and STRAUCH (1969); see also PAUL and TAKENO (1972).

In fact, IPATOVA et al. state the following. Let process C give a contribution 2WR J~, 1 to 2WR J~, then suppose that processes E and G, which are the same processes as C except for the higher order of perturbation theory, give a contribution 2WRJ~,2=t(2wRJ~,1)2. This suggests that the factor (30.36) is indeed of the form

1 1 (30.36')

The approximate equality accounts for an approximation made for the contribution from process C, which is restricted to the decay of the virtual local-mode excitation into the local mode itself and another phonon.

Another approach has the same kind of approximation and leads very directly to the exponential. One starts from an anharmonic Hamiltonian which is linear in all phonons except for the resonance phonon which appears in any arbitrary order. A canonical transformation of the Hamiltonian leads to the desired exponential. This was performed by HUGHES (1968) and also by IPATOV A et al. (1968, 1969 a, b). In terms of the self-energy of the resonance phonon this means, for example, that, of the internal lines in diagrams C and D of Fig. 30.1, all but one represent the resonance phonon. Within the framework of the approximate Hamiltonian, the above-mentioned authors are able to show that the total, main-line and sideband, absorption is a constant, as stated by (30.35). It seems that the method of canonical transformation cannot be applied to resonant modes because of the large number of vibrations whose frequency is close to the resonant-mode frequency and which participate in the resonant vibration to a more or less degree. The separation of the resonant mode from the other modes seems therefore impossible.


For local modes, the above argument may be a very good approach. For resonance modes, the use of just one diagonal element of the self-energy seems doubtful so that one resorts oneself to the eigenvalue treatment. From (30.33) the intensity is given in terms of the real part of

8 flA/8w2 =8 flH/8w2 - 8(flA- flH)!8w2

_8flH [ 8(flA-flH)/8w2] - 8w2 1 + 8flH/8w . (30.37)

The factor Re 8 flH/8w2 gives the harmonic intensity of the resonance line. The terms in square brackets then describe the anharmonic intensity reduction. The quantities have to be evaluated at the anharmonic resonance frequency; however, neglecting all derivatives of Re fl higher than the first, one may evaluate expression (30.37) at the harmonic resonance frequency as well.

To illustrate the result (30.37) one uses the definitions (30.14) for fl, (30.11) for n~, and (24.7) for m~. If one also uses 1+flA =0 for the resonance, one obtains for W2=W~

8 flA/8w 2 = 8 [(1 + go n~)-l go(v + Lln)]/8w2

= -g~m*[l-(m*)-l 8n*/8w2]. (30.38)

The term in square brackets corresponds to expression (30.31) for the Einstein oscillator. The main difference is the appearance of the effective mass which is a measure of the non-localization. The factor in front of the brackets is similar to the harmonic result,

8flH/8w 2 = -gom*,

and differs by a factor of (1 + go n6)' In addition, if the real part of 8 flA /8w 2 is taken, there is an additional term from the imaginary parts of g~ and the bracket. This term is small, since g~ and thus 1m g~ has to be small in order that a resonance rather than an antiresonance occurs. Due to anharmonicity the harmonic line intensity is thus reduced by a factor of approximately

1 +g~b6

8 (b*/m*)" 1- 8w2

The numerator tends to further decrease the line intensity, since generally g~ b < 0 for very low and very high frequencies. This is only true if processes A and B are sufficiently weak in the ideal crystal. The effect of the frequency dependence is the stronger the more localized the vibration, i.e. the smaller m* (m* > m) is. The neglected influence of the damping function and of the imaginary part of the Green function will be considered in connection with the anharmonic shift. Generally these terms tend to further decrease the intensity of a resonant mode, but not of a localized mode (where g~=O). Hence this term again takes account of the inertia of the lattice which adds to the "effective mass" of the oscillator.

The effect of temperature on resonant and local-mode line intensities has been observed in infrared-absorption experiments. The results are contradictory


Table 30.1. Temperature-dependent change in the intensity of infrared absorption lines due to reso-nant and localized modes

System T1(K) T2(K) I(T2)/I(T1) References

NaCl:Cu+ 2.3 79 0.7 WEBER and SIEBERT (1968) 2 80 0.60 ALEXANDER et al. (1970)

NaCl:Ag+ 0 300 -1 WEBER (1964), WEBER and NETTE (1966), WEBER and SIEBERT (1968)

NaI:H- 60 300 0.8 BAUERLE and FRITZ (1967) NaI:Cl- 1.4 26 0.3 CLAYMAN (1971) KCl:H- 77 295 0.55 SCHAEFER (1960)

100 300 0.8--0.75 MITSUISHI and YOSHINAGA (1962) 0 300 0.84 MITRA and SINGH (1966), 0 300 0.8 FRITZ (1968)

KCl:Na+ (pair) 2 42 0.6 TEMPLETON and CLAYMAN (1971) KBr:Li+ 2 30 0.15 ALEXANDER et al. (1970) KI:Ag+ 2 15 0.2 ALEXANDER et at (1970) CsI:Tl+ 11 77 -1 GENZEL et al. (1969) CaF2 :H- 20 290 0.75 ELLIOTT et al. (1965) SrF2 :H- 20 290 0.62 ELLIOTT et aI. (1965) BaF2 :H- 20 290 0.75 ELLIOTT et aI. (1965) CdTe:Be 4 290 0.4 HAYES and SPRAY (1969)

in many respects. Among other things, it was usually not clear how much of the spectrum was taken to belong to the resonance line and how much was considered to be background, multi-phonon, etc. absorption.

The intensity of the local-mode lines due to U centers in alkali halides will be discussed below. The high-frequency internal modes of molecular defects cause infrared absorption lines for which no intensity change at all could be observed (DEMYANENKO and TSYASHCHENKO, 1969; KONDILENKO et aI., 1970).

Most of the low-frequency resonant modes seem to show strong anharmonic effects; the corresponding lines shift and/or broaden considerably upon temperature variation. The intensity of most of the resonant-mode lines also decreases rapidly with increasing temperature. An example is shown in Fig. 30.3 a, taken from the extensive study by ALEXANDER et aI. (1970). Unlike most resonant modes, those of the system N aCI : Ag + and CsI: Tl + have a very weak intensity variation. Experimental data on this matter are given in Table 30.1. For results on AgBr:Li+ see HATTORI et aI. (1978).

Example: U centers in alkali halides The intensity of the local-mode lines due to U centers in alkali halides was

originally reported (SCHAEFER, 1960; PRICE and WILKINSON, 1960; MITRA and SINGH, 1966) to be strongly dependent on temperature. Later (MIRLIN and RESHINA, 1963, 1964b; FRITZ et aI., 1965; FRITZ, 1968) a weak effect was observed; the intensity was found to decrease typically by 10 % to 20 % upon heating the samples from 0 K to room temperature. In Fig. 20.2 above a typical absorption spectrum due to U centers at two different temperatures was shown. Within experimental error limits the total absorption is constant; with increas-

Sect. 3D

o

~ -0.1 u ~ -0.2 is' '3 -0..3

I

t:: -0.4 ::I

3.0.

2.6

"E 2.2 u

;::::: 1.8 '-l., N 1.4

, , , , , , , , , , , I

2IRR : ,0 I I I

I I

l , 0

Anharmonic effects in perturbed crystals 423

24

a

b

1.0

0.8

1m 1(0) 0.6

0.4

0.2

o 50 100 150 200 250 300 TOK

Fig. 30.4. Intensity variation of the local mode in KCl:H-. Circles: experiment (FRITZ, 1968). The curves

are explained in the text. (HUGHES, 1968)

... Fig.30.3a-c. Anharmonicity of the KI: Ag+ resonant mode. (a) Absorption-line intensity versus temperature. (b) and (c) Frequency shift and linewidth as a function of temperature. Circles: experimental data (ALEXANDER et ai., 1970.). Solid lines: smooth curves through experimental data. Dashed lines: approximate variation of the n;al and imaginary part of the

c self-energy

Temperature (OK)

ing temperature the intensity is shifted from the local mode to the sidebands. This is shown quantitatively in Fig. 30.4.

From the sum rule (30.35) and the relative sideband absorption a reduction of the local-mode intensity in KCI:H- by about 14% (90K) is found (FRITZ et al., 1965). From the slope of the damping function, Fig. 30.2b (77 K), the reduction is about 8 %. Higher-order corrections (ZEYHER and BILZ, 1969; Sect. 30k) scale the damping and shift function by a facor of about 2. This brings the reduction from J~ close to that from the sideband area.

Last but not least, one should refer again to the very thorough theoretical and numerical investigation carried out by HUGHES (1968), who takes various approaches to the general problem. As regards the comparison with the KCI:H- experimental data, (circles in Fig. 30.4) he calculates the Debye-Waller factor (30.36') from the sideband curves and arrives at the following/results, demonstrated in Fig. 30.4.

1) The Debye-Waller-factor (curve A) at the limit of high local-mode frequencies, OJL ~ OJ B , overestimates the intensity variation by a factor of about two, but there is some question about the interpretation of the experimental values, see below.


2) The leading term of the Debye-Waller-factor, i.e., expression (30.31) to first order, overestimates the effect even more (curve C).

3) Results of perturbation theory (where wL > w B but not necessarily wL

~ WB' W B a band-mode frequency, barely differ from the results for the low-order expansion of the exponential (curve B).

4) FRITZ' data seem to take into account the intensity increase of only, or mostly, the low-frequency sideband. However, the high-frequency sideband has the same absolute variation (see Sect. 30j below). Assuming the total absorption to be constant, FRITZ then underestimates the intensity variation of the main line by a factor of about two. Simulating this effect in the Debye-Waller-factor, HUGHES arrives at curve D which, in fact, is close to the experimental points.

5) HUGHES also calculates the intensity variation of a hypothetical resonant mode and finds that the approximation wJwB ~ ww'wB ~ 1 cannot be carried over to this case, as one would expect.

g) Anharmonic shift of resonance-mode frequencies. The anharmonic Einstein-oscillator frequency is directly obtained from (30.26) or (30.27 a),

wi - wi = 2WR LlRR (wR) or

%-wi=c5(O, oc; 0, ociwR)/m.

Since the shift function depends itself on the anharmonic frequency one expands it to first order and obtains from (30.26) and (30.27a)

wi -wi~2wRLlRR(wR)/[1-2wRJ~] =2WRJ R,

wi -wi ~(c5/m)/[1- o(c5/m)/ow2] iw2=W~ (30.39)

with a reduction factor similar to that connected with the width and intensity. In the general case one obtains the anharmonic shift from

(30.32) and

Re [1 + .uH(WR)] =0, (23.20)

where .u =.u~ is understood. In a similar expansion one has

1 + .uA(WR) = 1 + .uH(WR) + [.uA(WR)- .uH(WR)] +(wi _(2) O.uA/OW2iw2_w2 - R

or

(30.40)

In the linear case (for illustration) one obtains with 1 + g~ v = 0 for w2 = wi from (30.11)

(30.41)

Restricting oneself to lowest-order effects one can evaluate the denominator in (30.40) at w2 = wi instead at w2 = wi and obtains with (30.38)

_ Re egA n*] c5* /m* w2 _ w2 - 0 '" -'-------::-:-::-:---'---""'--'-::--0;-

R R-Re[g~m*(1-m*-lon*/ow2)] "'1-o(c5*/m*)/ow2· (30.42)


Again, the anharmonic shift due to its own frequency dependence is reduced. Also, the non-Einstein behavior of the resonance involves 15* /m* rather than 15* /m (> 15* /m*), which further reduces the effect. The frequency dependence of the damping function has been neglected, as in Sect. 30e. The approximate results (30.26) or (30.27 a) will always overestimate the actual shift. For lowfrequency resonances one often finds a positive anharmonic shift due to the process A. In this case the neglected term in the numerator of (30,42) supports the anharmonic shift due to 15*.

It is probably pointless to actually compute the different quantities which appear in expression (30.42), since the approximations may turn out to be too crude. In this case one would have to resort to the exactly calculated spectrum and find the shift a posteriori. In the following part of this section the influence of the various processes which contribute to the unreduced shift will be investigated.

All four terms of (30.20) or (30.20') contribute to the shift of normal modes. The first two terms (diagrams A and B) exhibit the same temperature

dependence due to their common factor 1 + 2 n A which for very low and very high temperatures can be approximated by

1 +2 exp( -hwJkT)~ 1 for kT ~hWA

1+2n..l~ kT/hw..l for kT~hwA'

(30,43 a)

(30,43 b)

Processes A and B contribute a shift which is proportional to TO and Tl at very low and very high temperatures, respectively.

The Green function (30.22) or (30.22'), involved in process C, has two parts. The first (upper sign, "summation process") has a factor very similar to 1 +2n, the other (lower sign; "difference process") vanishes at vanishing temperature, while at high temperatures it is proportional to T, as is the first part.

Similar arguments can be given for the shift produced by process D, with the result that the contribution to the shift is proportional to T2 at high temperatures.

As for the sign of the shift, consider a defect in an alkali halide vibrating in a resonance mode (A = R) in its extreme case of an Einstein-oscillator mode. Since in this limiting case only the defect vibrates in the resonance mode, the arguments in the self-energy, (30.20') are (/; l') = (L, r:t.; Z;, r:t.') = (000, r:t.; 000, r:t.'). Born-Mayer and Coulomb potentials probably give a reasonable description of the interionic potential in alkali halides. It is then sufficient to neglect m =(M, y) referring to neighbors beyond the nearest ones. For a defect with 0h symmetry, one also has r:t.=r:t.'. The potential expansion terms which are involved in process A are then

4>(0, r:t.; 0, r:t.; 0, [3; 0, [3) and 4>(0, r:t.; 0, r:t.; M, [3; M [3).

Now the fourth-order derivatives usually predominate over the third-order terms; then only the expansion coefficients with r:t. = [3 are needed. The expansion coefficients from a Born-Mayer potential are generally larger than those of a Coulomb potential. Since the former are positive one finds that the shift due to process A is positive. The occurrence of terms involving, e.g., 4>(0, r:t.; 0, r:t.; 0, r:t.;


M, IX), which might give negative contributions, is completely ruled out for the Einstein-oscillator case.

In addition to the approximations made by assuming an Einstein-oscillator resonance and two-body, nearest-neighbor, and central interactions one has assumed that the defect-lattice short-range interaction, even if not of BornMayer form, at least has a fourth-order derivative which is larger than the corresponding one of the Coulomb potential. This latter argument led to the conclusion that the shift due to process A is positive. To which extent the argument is true cannot easily be deduced from experiment, since there are more processes causing a shift. Whenever there is a term proportional to V4 there is a competing one with (V3)2.

Here the competing process is the one represented by diagram B of Fig. 30.1. Since in this process the third-order anharmonicity appears essentially squared, the resulting shift is expected to be negative due to the sign in front of the corresponding expression in (30.20). This is, in fact, true if one considers anharmonic interaction between the defect and its neighbors only, and the resulting shift of an Einstein-oscillator frequency.

For the investigation of the various contributions from band and resonance phonons, analogous arguments can be made for the phonons represented by the "balloon" of diagram B, (30.24), and for the "string" holding the "balloon". It should be noted that, e.g. in crystals with inversion symmetry, the "string" phonons are restricted to being even-parity phonons and cannot be resonance modes with the motion of the defect involved. A more careful analysis seems to be necessary, though. In general, an explicit investigation of the self-energy is necessary. The only investigation of this kind we know of is that by TIMMESFELD (1968). Unfortunately TrMMESFELD'S results for the shift from process B contain a sign error.

The sign of the shifts due to the processes C and D depends only upon the sign of the Green functions (30.22), (30.23), since the coupling parameters come in squared. At first sight the Green functions seem to be negative (positive) for high (low)-frequency resonances resulting in a positive (negative) shift. However, care has to be used, if one or more of the quantum numbers in the Green functions refer to the resonance vibration itself. For a low-frequency resonant mode the conclusions are unaltered, but if some of the quantum numbers in the Green functions refer to a localized mode, then the resulting shift may alter the sign. As an example consider process C for an Einstein localized mode. This mode may virtually decay into two band modes, into a band mode and a local mode, and into two local modes (if allowed by symmetry). Expand the Green function in terms of WB/WL, the ratio of band- and local-phonon frequencies, If one keeps the leading terms, one obtains for the two-phonon Green function at frequency wL

G(L, IX; £, IX'IM, fJ; M', fJ')

= "-- I X(L, IXI A) X*(£, IX' I A)x(M, fJl X) X*(M', fJ'l X) gu' (30.44) u'


with

With every factor gu' is connected a product of eigenvectors, X(A)xt (A)x(A,)xt (A'). Let the band frequencies be approximated by an average, wB and the eigenvectors by

L X(A) X t(A) ocM- 1 ~w~/I[>, A

then G is estimated as

(w )3 1 I G~WL [-(1+2nB) W: +2(1+nB)+3"(1+2nd ] 1[>2. (30.45)

At low as well as high temperatures the first term is small compared with the other two. The main contributions to the shift from process C for a localized mode are, therefore, in this approximation caused by a virtual decay into phonons of which at least one is the local-mode phonon itself. The resulting shift is then negative again. For process D the arguments follow the same line and lead to the same results.

In summary, all processes except process A are expected to cause a negative shift for low- as well as high-frequency resonances. The approximations made for the shift from the frequency-dependent processes C and D may be very crude. In particular, the virtual decay of a local mode into two band phonons may be strongly enhanced when the local-mode frequency is closed to the two-phonon cut-off frequency 2wmax ' In fact, the localized-mode frequency of a D- V-center is generally closer to 2 W max than the H - frequency is. Consequently, the shift of a D - frequency is greater than that of the H - frequency in the same host crystal, as has frequently been found (ELLIOTT et aI., 1965; FRITZ et al., 1965; DOTSCH et aI., 1965; DOTSCH, 1969a; DOTSCH and MITRA, 1969; OLSON and LYNCH, 1971), see Fig. 30.5 and the discussion below.

Thermal expansion is treated in Sect. 31 in the connection with strain (see also Sect. 29c). It turns out that the shift due to thermal expansion can be estimated from the strain-induced frequency shift (Sects. 29c, 31) since in both cases the shift is caused by the variation of the lattice constant. Figure 30.3 b shows a resonant-mode frequency shift as a function of temperature. Figure 30.5 shows the frequency shift of H- and D- induced local modes. The shift from thermal expansion is included in the figures. As stated above, the shift of the Dmode is greater than that of the H - mode. The net shift for both defects is greater than that due to thermal expansion. This indicates that the first term in (30.44) is larger than the other two (in which case the approximation (30.45) is far too crude) or that process A contributes a larger shift than the other processes (C and D etc.) or both.


KCI

~:=: ~= } expo

10 Co---o difference b-a d --- quasi harmonic

shift (extrapolated 1

----7''-+-----i 10

-15 '------'------'->-----::--'---",.-----'-15 o

Sect. 30

Fig. 30.5. Frequency shift of the local mode in KCl: H- and KCl: D-. (BILZ et aI., 1966)

For resonant modes with very low frequencies, a pOSItIve shift has been observed in a few cases (GENZEL et aI., 1969; PRETTL and SIEP, 1971 a, b, c; CLA YMAN, 1971). In view of the fact that there is always the negative contribution to the shift as well, the net positive shift of the low-frequency modes indicates a very strong bilinear anharmonicity. For the KI: Ag+ resonant mode, ALEXANDER et aI. (1970) estimate the thermal-expansion-induced shift to be a small fraction of the total shift, see Fig. 30.3 b.

Since the U-center local-mode intensity varies little with temperature the frequency shift reveals directly the real part of the self-energy. This is not so in the KI: Ag+ resonance. If one identifies the experimentally observed shift with the approximate shift JR , (30.39), one can calculate the real part of the selfenergy, ARR, using the experimentally observed variation of the factor 1-2wRJ~ as taken from Fig. 30.3a. Since ARR(T=O) and ARR(T) cannot be found separately from Fig. 30.3 b, the so calculated variation of ARR with temperature is probably overestimated. The fact remains that ARR varies more strongly with temperature than does JR.

h) Isotope effects. In this section the effects of the replacement of a defect by another isotope will be studied. In the harmonic case PAGE (1974) and PAGE

and HELLIWELL (1975) find that the isotope splitting is always less than the


Einstein oscillator splitting,

(30.46)

where W 1 and W 2 are resonance frequencies of two isotopes with masses m1 and m2 (m 2 >m 1 ), respectively; mt are the respective effective masses, cf. (24.7), which include some of the inertia of the lattice. Relation (30.46) seems generally fulfilled except for the 6Li and 7Li isotopes in KBr and 35Cl and 37Cl isotopes in NaI (KIRBY et al., 1968b; CLAYMAN et al., 1969), where anharmonic effects are thought to be responsible for this "negative" isotope shift (PANDEY et al., 1974).

PAGE's argument can be summarized as follows. Given a resonance frequency W 1 by

[1 +,u'l(W 1 )] l/tl =0

and taking the derivative with respect to a variation in the defect mass (and thus the resonance frequency),

d ,8¢ 8¢ 8w2

0=dm(1+,u)=8m +8w2 8m

one finds with (24.7 a) and v 1 = LJ f - w2 LJ m 1

8w2 8,u'/8m

8m 8,u'/8w2

l/t1 v(8 g~ v/8m) l/t 1

l/t1 v(8 g~ v/8w 2 ) l/t 1

l/t1wil/tl l/t1 mi l/t 1 •

(30.47)

To first order, one finds (30.46); also, only in the Einstein-oscillator case there is no inequality in (30.46). It is now clear that any splitting larger than the isotope splitting has to be explained by anharmonic shifts as long as the first order of the perturbation above suffices.

The model of SIEVERS and TAKENO (1965) predicts an isotope splitting which is larger than that of Einstein oscillators. However, SIEVERS and TAKENO'S expression for the splitting contains terms of different orders in the small parameters (LJm 1 -LJmZ ' 8g~/8w2), as PAGE (1974) points out.

KLEIN (1968a, b) defines a further effective mass. He starts out from the defect-site element 1+go(LJep-w2 LJm) of l+gov and interprets all the other terms of det(1+gov)=1+go(LJep-w z LJm**) as an effective mass. The conclusion is, as expected, that with increasing size of the impurity space the inertia of the lattice leads to an increasing effective mass of the defect. We will not follow this type of effective mass any further here and refer the reader to KLEIN's articles.

Generally the fit of the observed isotope splitting in a harmonic model is unsuccessful, even if parameters in a more extended impurity space are included (MACDoNALD, 1966; BENEDEK and NARDELLI, 1967b; BENEDEK and MARADUDIN, 1968; BENEDEK, 1970; KIRBY et al., 1968b). This is a clear indication that anharmonic effects have to be taken into account.


In the anharmonic case one starts from (30.13). The poles of gA are given by

Re det [1 + go(v +7t*)J =0;

neglecting small contributions from imaginary parts of go and 7t* = 1)* - i r* one finds similarly

oro2 I/It (ro2 -01)*/om)I/I om = - I/It(m* -(1)*/oro2 ) 1/1 . (30.48)

At high temperatures the contribution from processes A, B, C and D to (1)* /om can be shown to vanish. For example, the mass dependence of processes A and B is containedjn the factor (30.24), which becomes

1 h 00

i<v(l) v(l') =2:;- Joo

dro(l + n",) 1m G(l, 1'1 ro)

- ikT ~ 7 dro 1m G(l, 1'1 ro)/ro =i k T Re G(l, 1'10) (T-oo) 7t -00

=ikT[<D-1Jll" (30.49)

cf. Sect. 24a. The result is the real part of the Green function at zero frequency which depends on force constants only, but not on masses.

For T=O K and for an Einstein oscillator, expression (30.24) becomes

i<v(O) v(O) = h/8mro r och/8 vim and

£5 = hai < v (0) v(O)

where a is the fourth-order potential expansion term. Then

0£5/oro2 =0, o£5/om<O.

The effect is to enhance the value of loro2/oml if a is positive (as from BornMayer potentials), and the same is true if the restriction to an Einstein oscillator is relaxed. By similar arguments, process B has the opposite effect of process A, and for low-frequency resonances process C has also the opposite effect of process A.

F or the anomalous isotope splitting to occur, as in KBr: Li +, there are the following prerequisites: Firstly, process A has to predominate over all the other processes which contribute to the shift. This generally means that the fourthorder anharmonicity has to be very strong. Secondly, the fourth-order anharmonicity has to have a positive sign, i.e. the short-range potential coefficients have to predominate over those of the Coulomb potential. Thus, good candidates for an anomalous isotope splitting are all those. defects whose resonance frequency show a shift to higher energies with increasing temperature. Thirdly, the contribution of the resonance mode to the thermal phonon displacements, the "balloon", has to be sufficiently strong. This seems to support light defects as good systems. The light Li + isotopes in alkali halides, where they also exhibit low-frequency resonant modes, are ideal in this respect. Finally, the temperature


has to be sufficiently low, since at higher temperatures the effect tends to vanish, because of the additional factor from the phonon occupation number, see (30.43 b). For results on NaI:Cl- see STRAUCH and PAGE (1978).

Isotope effects on the anharmonic width will be considered in Sect. 30i.

i) Anharmonic width of resonances. In Sect. 30e it has been shown that the frequency dependence of the resonance-mode shift function causes a reduction in the anharmonic width of the resonance. This is due to the fact that for a negative slope of the shift function and for a frequency just off the resonance, the frequency appears to be shifted away from the frequency under consideration. Thus the width of a resonance has to be analyzed in terms of both the imaginary part of the self-energy, r, and the reduction due to shift function, (30.30) for the Einstein resonance and (30.34) in general. In this linear case, one obtains from the latter equation

1m {g~(v+ LI n)}

Re g~ [LI y + (Re g~) -1 (1m g~/m*) (Re g~) -1] Re {g~m*[I-(m*)-1 on*/ow2]}

(Rego)-1 (1m go/m*)(Re go)-1 + y* ~ 1-(m*)-1 o{)*/ow2 (30.50)

Terms of second order in both, g~ and Y6, have been neglected. In this approximation the first term in the numerator gives the harmonic width, (24.13); the second term is the anharmonic width; these superpose, and the total width is reduced, as is known from the Einstein case.

It is quite possible that the increase in width with temperature is largely compensated by the increase in the reduction factor. An over-compensation, i.e. a line narrowing with increasing temperature, has to our knowledge not so far been observed in infrared or Raman spectra. It appears, though, that from intensity variation and shift the KI: Ag+ resonance has a larger anharmonicity than the NaCI: Cu + resonance (ALEXANDER et aI., 1970). Yet the experimental width is observed to be about the same in both systems. This might be, at least partially, due to the reduction from the frequency dependence of the shift function.

The reduction of the width is exactly like that of the intensity and the shift, as discussed above. We will therefore consider now the imaginary part of the self-energy which gives the unreduced anharmonic width.

Processes C and D (not A and B) depend on frequency, whence they have imaginary parts. Like the shifts, they are constant near T=O K and at high temperatures are proportional to T and T2, respectively. For the following argument, it is useful to distinguish between decay (summation) and scattering (difference) processes (see Sect. 30g). For resonances of very high and very low frequency, the summation processes do not contribute: For very high-frequency localized modes there are no two or three band phonons into which the localized phonon can decay under conservation of energy. For very low-frequency resonant modes (in the limit of the resonant frequency going toward


zero), the resonant phonons decay into phonons with even lower energy, the density of which decreases with decreasing resonance frequency. In reality, of course, the resonance frequencies are not infinitely small or large. There are always at least a few phonons into which a low-frequency resonance can decay and even the rather high-frequency V-center localized mode can decay into four phonons, even if in some cases not into three and in many cases not into two. As a matter of fact, at T = 0 K the scattering processes are zero (because there is no phonon that can be annihilated) and the width is due merely to decay processes (apart from the "harmonic" width of a low-frequency resonant mode).

At higher temperatures decay and scattering processes have the same temperature dependence and therefore are equally important as far as their temperature dependence is concerned. The number of phonons whose energy difference is equal to the small frequency of a resonant mode may be much larger than the number of phonons whose energy sum is equally small. We conclude that above certain temperatures difference processes are more important than decay processes for low-frequency resonances. This has been numerically verified by TIMMESFELD (1968).

For high-frequency localized modes that part of process C in which three band phonons are scattered is not allowed because energy cannot be conserved. However, there is the specific scattering part of process D in which a phonon is scattered at the resonance phonon, that is, the local mode and another phonon are annihilated and created again. This process is allowed at finite temperatures for resonance modes of any frequency. For a Debye model it goes as T7 and T2 at low and high temperatures, respectively.

The experimentally determined temperature dependence of the KI: Ag+ width is shown in Fig. 30.3 c. The situation here is similar to the anharmonic frequency shift. If the experimental width is identified with 2 FR, (30.30), the imaginary part of the self-energy, rRR , can be calculated and is included in Fig. 30.3 c. A very drastic change of the self-energy with temperature is the result.

The separation of the harmonic width from the anharmonic width at T = 0 K presents a problem. On the other hand, the relation between FR and rRR is an approximation, anyway, valid only for a weak frequency variation of rRR(W) and small curvature of LlRR(W). It has to be pointed out, however, that the imaginary part of the self-energy is expected to be closer to the dashed curves in Figs. 30.3 band c than to the full curves.

Example: V centers

From the weak intensity vanatlOn of the V-center local-mode lines it is appearent that the width is only weakly reduced and thus displays more or less directly the imaginary part of the self-energy. Experimental data from V centers in KBr (FRITZ et aI., 1965) are shown in Figs. 30.6 and 30.7. Figure 30.7 contains in addition the results of calculations performed by GETHINS (1970). The variation of the width with temperature is as follows.

At very low temperatures the width is determined only by decay processes which contribute a constant width at low temperature. This is found both experimentally and theoretically. The decay of the D - local mode into two band phonons is possible while the H - mode has to decay into three band modes

Sect. 30

80

50 cm-1

20

Anharmonic effects in perturbed crystals

10 20 50 100 200

o •

10~-~------------~r_~-~--~

C

:t: 2

0.5 f----II'f---c

0.2

0.1 '--_-'--_-'-lL-_--l-_-'--_-L_---''--' 5 10 20 50 100 200

Temperature

433

Fig.30.6. Half-width of the local mode in KCl: H- and KCl: D- versus temperature. Squares, experimental data (FRITZ et aI., 1965). The curves are explained in the text. (BILZ et aI., 1966)

in order that energy is conserved. However, Gethins finds that the D - width from decay into three phonons is an order of magnitude larger than from a decay into two. The width from the decay of the D - mode into two (three) phonons is proportional to the projected two- (three-) phonon density, i.e., the two-(three-)phonon Green function. Since the D- mode frequency is not too far from the two-phonon cut-off, the two-phonon density is small at the D- mode frequency, while the three-phonon density is much larger. There is thus a large width resulting from the three-phonon decay. On the other hand, the decay of the H - local mode into three band modes is less probable than the decay of the D - mode since the H - mode frequency is close to the three-phonon cut-off. The H- width at T=OK is then much smaller than the D- width. In fact, the width of the H - line has not yet been resolved experimentally (Fig. 30.6). With increasing temperature the scattering from process D becomes increasingly important. In the Debye model this process goes as T7 and T2 at low and high temperatures respectively. Experimentally a T3 behavior at the resolution-limited lowest temperature is observed in KCI (Fig. 30.6) and a T 5 behavior is calculated for KBr. At higher temperatures the width approaches a T2 behavior. An analysis of the width caused by the scattering process D shows that


Ioor---------------------------~~ .. KBr: KD theory .;/

SD • KBr. KH theory e"". " ;/ ,,' , ... KBr· KD expt.(o) ,r / '~

ID .. KBr· KH expt.(a) ." ?:, '. , · "d" n~.(') / •• ~

~: .. 0_-; ;y :z: D·I -I {/

0.05 xlOO

O'OIl-1 --'---'--'-S ... /.~I~O.---1--L....1..f.So.... .L..I..';IO~O.---'--'-~SOO TEMPERATURE (K)

Sect. 30

Fig.30.7. Calculated values of the half-width of the local mode in KBr: H- and KBr: D- are shown as a function of temperature in a log-log plot, together with experimental results of (a) MIRLIN and RESHINA (1964) and (b) FRITZ et al. (1965). Solid lines have been drawn through the theoretical points, dashed lines through the experimental ones. Note the change of scale. (GETHINS,

1970)

y* is independent of the defect mass at high temperatures. The ratio of the widths for two isotopes is thus, from (30.50),

y~ _ yT/mi _ (mi)-l YA -y*/m* - m* .

2 2 2 2

This is what is found both experimentally and theoretically. Figure 30.6 shows that at high temperatures the D- line width (curve c) is one half of the H- width (curve a). At lower temperatures decay processes become important for the D - mode, and the line widths are no longer proportional to one another. Curve c in Fig. 30.6 is calculated from the scattering part (analogous to curve a) and a two-phonon decay process (estimated as curve b). The superposition of these two gives curve c which, in fact, is a very good reproduction of the experimental data.

Lack of experimental data prevents a complete test of GETHINS' theory. At low temperatures the D - width is of the same order of magnitude in KBr (theoretical) and KCI (experimental). At higher temperatures the theoretical width (Fig. 30.7) is too small by a factor of about 10. GETHINS feels that an error of a factor of 3 in the determination of the fourth-order anharmonicity coefficient is unlikely. He points out the importance of process E (Fig. 30.1) and possibly higher-order decay and scattering processes.

Care is also essential in determining the width at T = 0 K, since the residual width may depend on the defect concentration. The observations of the BaF2 : H - local-mode linewidth as a function of temperature and concentration of defects are shown in Figs. 30.8 and 30.9.

j) Multi-phonon spectra, sidebands, overtones. The anharmonic properties described above are properties of the damping function evaluated at the resonance frequency wR • At frequencies co sufficiently far away from the reso-


15

10 Ba~: Wfundamental

o 4,4xl02Ocm- 3

t Do 9,Oxl019cm-3

5 x 9,4 xl018cm-3

IE slope=1.S

~ E "0

2 Do

'~ Do Q)

C ::::;

x

10 20 50 Temperature (K)-

Fig.30.S. Linewidth of the fundamental local mode in BaF2 : H - as a function of temperature for three H - ion concentrations, (HARRINGTON and WALKER, 1971)

5,0 BaFz: W fundamental

t 1~2.0 yf

JloyH 0,5

0,1

o Raman

A Infrared

0,2 0.5 2

Concentrati on (1020cm-3) ---

5

Fig.30.9. Residuallinewidth of the fundamental local mode in BaF2 :H- as a function ofH- ion concentration, (HARRINGTON and WALKER, 1971)

nance frequency the imaginary part of the anharmonic Einstein-oscillator Green function in momentum representation can be approximated by

(30.51)

More generally one has

(30.52)

where the eVA are the solutions of

(W 2 _W2 )

det ;wA

bu,+Llu(w) =0 (30,53)


which are, one hopes, close to the solutions of

However, it seems impossible to find the solutions of (30.53) for reasons similar to the ones encountered in Sect. 22c, i.e. because of the dimensionality of the problem. This was discussed at length in Sect. 30c. Integrating expression (30.51) over all infrared frequencies gives

which was used in Sect. 30e. Together with the results of Sect. 30f this means that to first order in J~ the integrated one- plus multi-phonon spectrum, (30.36) and (30.54), is a constant, as postulated by more general arguments, (30.35). See also Sect. 30 f.

For the practical determination of J~, a clear separation of the resonance line from the multi-phonon spectrum is necessary. This will generally be given if rRR(W) is small and has a minimum around w=wR as is needed for the validity of (30.29), cf. Fig. 30.2.

If one considers process C as contributing to the shift function, one can see that there are three types of contributions, as in expression (30.44). The first describes the excitation of two band phonons and corresponds to the twophonon excitation of the pure crystal. However, the band phonons may be perturbed, etc. as discussed in Sect. 30 g. The second contribution describes the excitation of a resonance phonon and a band phonon. In absorption spectra this is seen as a sideband accompanying the main resonance line. The final contribution describes the excitation of two resonance phonons and thus leads to absorption due to the first overtone. Process D is analogously discussed.

Due to the crystal symmetry the first excited (overtone) level of a resonance or localized oscillator is split. Overtone bands are subject to strict selection rules; in fact, the number of overtone lines of a given order can be used to determine the symmetry of an impurity site. This field is reviewed by NEWMAN (1969). For the estimate of the intensity of overtones, see MARADUDIN and PERETTI (1967). For transitions between the different sublevels of an overtone, see PINKEVICH (1977).

In alkali halides the V center seems to be the only defect which produces a local mode; overtones can be observed by infrared absorption techniques in the rare-earth halides (ELLIOTT et al., 1965; JONES and SATTEN, 1966; HAYES and MACDONALD, 1967; HAYES and SPRAY, 1969; HARRINGTON et aI., 1970). Recently, a local mode in doped silver halides has been observed (HATTORI et aI., 1973, 1975) including the second overtone in AgBr: Li +. In the covalent crystals local-mode overtones are more frequent (YUASA et aI., 1970; MANABE et aI., 1971; NEWMAN, 1969; with references prior to 1969). Recently, Raman scattering by V-center local-mode overtones was observed (MONTGOMERY et aI., 1972b; WOLFRAM et aI., 1972).


50r---------------~------------~

--1 I-40

NaCI:Cu+

(3x10 18/cm 3 )

"'30 .......

'" C :::J o

U

20

0---<> 6.3 K f::, ____ f::, 17.8 K

10 0''''''''0 25.8 K

oL~~~--~~--~~~ o 60 Freq uency shift (cm-1)

437

Fig. 30.10. z(x x) y Raman spectrum of NaCl: Cu + for several different temperatures. The instrum· ental resolution is 3 cm -1. The symmetries of the two scattering peaks are indicated. (MONT,

GOMERY and KIRBY, 1971)

Evidence for overtones of resonant modes have also been found (MONTGOMERY and KIRBY, 1971; GANGULY et aI., 1972). The Raman spectrum of NaCI:Cu+ (Fig. 30.10) shows strong r;i and r1+ peaks at 38.6 and 46.5 cm-l, respectively. These were first interpreted as one-phonon resonances. They also show up as sidebands of the r 15 resonance (at 23.5 cm -1) at a separation of 40.9 and 46.4 cm -1, see Fig. 30.11. (The small peak at 59.2 cm -1 has been attributed to a resonant mode due to unwanted F - impurities). Recently, these "resonances" have been interpreted as overtones of the r 15 resonance. This is supported by the fact that the Raman intensity variation with temperature is stronger than 1 + n(wR), as it would be for a one-phonon resonance.

Sidebands of the U-center local mode will be discussed below. Sidebands have been reported in CdTe:Be (SENNETT et aI., 1969); GaP:B (HAYES et aI., 1969; contradicted by THOMPSON and NEWMAN, 1971); KI:Ag+ (KIRBY, 1971a, b); CdS:Be and CdSe:Be (MANABE etal., 1973); AgBr:Na+ (HATTORI et aI., 1975); and probably others. Resonant-mode sidebands have been calculated by BENEDEK and NARDELLI (1966b) and BENEDEK (1971).

Finally, sidebands, overtones and combination bands are observed in defect systems with internal degrees of freedom. For these reference is made to Sect. 20g.

The frequency dependence of w~ = w~ + 2WR LlRR causes the overtone frequency to be different from the multiple of the fundamental frequency. This, too,

438

'E u

80


NoCI: Cu+(250ppm) 4.2 K

+

r- X100

0L-~~=d==~~=I~~-L~-L~ o 20 40 60 80 100

Frequency (cm-1)

Sect. 30

Fig.30.11. Far-infrared absorption spectrum of NaCI:Cu+. Note the changes in the absorption coefficient scale by factors of 10 and 100. The peak at 23.5 cm -1 is the fundamental absorption of the resonant mode; the peaks at 64.4 and 69.9 cm -1 are second harmonics of this mode. The peak

at S9.2cm-1 is attributed to unwanted F- impurities. (MONTGOMERY and KIRBY, 1971)

is often seen in the sideband spectrum and therefore should not be neglected. This problem will be taken up below. In this connection one should mention that the multi-phonon spectra in this picture arise through virtual excitation of the resonance oscillator. There are other virtual excitations, for example, electronic transitions, which induce multi-phonon spectra. In infrared absorption the latter example is often formally interpreted in terms of absorption by higherorder dipole moments, see Sect. 27 e. It should be noticed that this kind of multiphonon spectrum integrated over the infrared frequency region is not independent of the temperature, unlike the absorption due to first-order dipole moments as discussed above. If one includes real electronic transition lines in the integrated absorption spectrum, the latter is expected to be independent of the temperature again. As long as one confines oneself to infrared frequencies, the amount of non-constancy of the integrated absorption is a measure of the absorption due to non-linear dipole moments (STRAUCH, 1969).

Example: V -center local-mode sidebands

Sidebands were first observed in V-center doped alkali halides by FRITZ (1962, 1965); FRITZ et ai. (1965); with subsequent investigations by numerous authors both experimental (DOTSCH et aI., 1965 (LiF, NaF); MIRLIN et aI., 1965 (KCI); MITRA and BRADA, 1965 (KCI); ELLIOTT et al., 1965 (CaFz' SrFz, BaFz); MITRA and BRADA, 1966 (CsBr); TIMUSK and KLEIN, 1966 (KBr); BAUERLE and FRITZ, 1967 (NaI), 1968b (KI); FRITZ et aI., 1968 (KI); DOTSCH, 1969a (LiF, NaF); DOTSCH and MITRA, 1969 (CsCI, CsBr, CsI); DORR and BAUERLE, 1970 (KBr:Hn; HARRINGTON and WALKER, 1970 (CaFz); MACPHERSON and nMUSK, 1970a (NaF, NaCl, NaBr, KCI, KI); OLSON and LYNCH, 1971 (CsBr, CsI); and HARRINGTON et aI., 1971 (BaFz)) and theoretical (ELLIOTT et aI., 1965; TIMUSK and KLEIN, 1966; BILZ et al., 1966; XINH, 1966, 1967a; PAGE and DICK, 1967; GETHINS et aI., 1967; BILZ et aI., 1967; KUHNER and WAGNER,


1967; STRAUCH and PAGE, 1968; WAGNER, 1968a; ZEYHER and BILZ, 1968, 1969; BOESE and WAGNER, 1970; MACPHERSON and TIMUSK, 1970a; BLUTHARDT et aI., 1973). The pressure dependence has been investigated by SHOTTS and SIEVERS (1973).

The local-mode sidebands arise due to simultaneous excitation of the local mode (quantum number L) and a band mode (2). Generally this can be accomplished either by direct excitation via the second-order dipole moment (M 2) or via virtual excitation of other phonons (2'). The expression for the dielectric susceptibility in the local-mode sideband frequency region can be given in an approximation of (9.14) as

x"iw)cx: IIM2,,(L, 2)-I Mtcz(2') gl(2') V3(2', L, 2W g2(L, 2) . .l. .l.'

The two (interfering) processes have the same temperature dependence (from g2) and thus cannot be distinguished from temperature effects on the sidebands. The summation and difference bands vary as 1 + nL + n.l. ~ 1 + n.l. and n.l. - nL ~ n.l. respectively. At low temperatures n.l. is small. The variation of the sidebands of the local mode in KBr:H- (at 447 cm- I ) is demonstrated in Fig. 30.12. On the other hand, the temperature independence of the total integrated local-mode main-line plus sideband spectrum indicates that contributions from the seondorder dipole moment are negligible. This conclusion, however, carries a sizable experimental error. More conclusive is the frequency dependence of the sideband spectrum, since the virtual phonon propagator brings in an extra

CIl .2

---Wavelength {ILl 28 26 24 22 20 18

1.0 ..-'=r--=r-r-~-TTT----"r'-----TT----,--T-----='r----, KBr: HI =300K 2= 90K 3= 55K 4= 21K 5= 15K

C 0.11----+--r-t-++-+-~-+_--_+---+_--_+--__i c ti c: o u c: o ~ ... o en .a cO.Ol~-~~+---+-+---+---++~~-~~--+----4 CIJ > ~ a; 0:::

350 400 450 500 550 600 650 700 Wavenumber {em-II_

Fig. 30.12. Absorption spectrum of H - centers in KBr. The absorption constant is given on a logarithmic scale. (FRITZ et ai., 1965) .


frequency factor

The most important of the intermediate phonons is the local mode itself, for which the frequency denominator becomes especially small,

with OJ' = OJ - OJL being the frequency splitting from the local-mode frequency. In general all phonons may couple to the local mode. In alkali halides

inversion symmetry about the defect allows only even-parity phonons to couple. Assuming anharmonic coupling between the defect and only nearest neighbors further reduces the allowed phonons to those of A 1g, Eg, and Tzg (I~+, I;.i, and I;~) symmetry. Assuming central potentials between the defect and its nearest neighbors leaves one with only two anharmonic coupling coefficients, A =<P(O,x; O,x; 100,x) and B= <P(O,x; 0, x; 010, y)=<P(O,x; O,y; 100,y). Analogous expressions are obtained for the second-order dipole moment coupling coefficients. Estimating A and B from Coulomb and Born-Mayer potentials gives I B I < I A I and BA < O. The contributions from the three symmetries to the sideband spectra are weighted by (A+2B)Z, (A-Bf, and B Z, whence one can expect the Eg (I;.i) spectra to be most important. Thus the sideband spectrum becomes a complicated function of second-order dipole moment, anharmonic and harmonic coupling constants; still it seems that it has been largely unravelled.

The sideband spectrum as experimentally observed, see e.g. curve a of Fig. 30.13, usually vanishes well below the one-phonon cutoff. In the acoustical-phonon frequency region phonons of mainly r1i and I;~ symmetry give contributions. Assuming a nearest-neighbor central-force-constant change, as is necessary to reproduce the local-mode frequency, the phonons of 1;.+ and I;.i are perturbed while those of I;~ are not. PAGE and DICK (1967) have done a rather extensive comparison of the various contributions with the experimental curve in KBr:H-. They agree with TIMUSK and KLEIN (1966) that anharmonic coupling seems to give a better fit than that due to second-order dipole moment, even though the agreement is still poor, Fig.30.l3b. The coupling parameter A seems to predominate over B, as anticipated.

The rather strong force-constant weakening of about 50 % needed to reproduce the local-mode frequency is expected to be accompanied by lattice distortions and thus by force-constant changes to further-out neighbors. This led GETHINS et al. (1967) to allow for force-constant changes between nearest and fourth-nearest neighbors (i.e. along the cubic axes). This improves the agreement greatly, Fig. 30.13 c. Attempts have been made is diminish the discrepancies which still remain by allowing for a larger coupling coefficient B (KOHNER and WAGNER, 1967; WAGNER, 1968). This has been shown (STRAUCH and PAGE, 1968) to lead to worse agreement in other crystals. Instead, a nearest neighbor non-central force-constant change has been suggested. The inclusion of this parameter leads to the theoretical curve in Fig. 30.13 d.


KBr:H-

a TKexp

d

SP

BZW

o cm-1 150

Fig.30.13a-f. Experimental and theoretical sideband spectra in KBr: H-. (a) Experimental (nMUSK and KLEIN, 1966); (b) A=FO, Llilll =FO (nMUSK and KLEIN, 1966); (c) A=FO, Llilll =FO, Lli211 =FO (GETHINS et a!., 1967); (d) A =FO, B= C=FO, Llilll =FO, Lli211 =FO (KUHNER and WAGNER, 1967, WAGNER, 1968a); (e) A=FO, Llilll =FO, Lliu. =FO, Lli211 =FO (STRAUCH and PAGE, 1968); (f) higher-order effects eliminated from experimental spectrum, see text (BILZ et a!., 1967, ZEYHER and BILZ, 1968,

1969)


A shell-model treatment of the dynamics of the perturbed lattice has also been employed (PAGE and STRAUCH, 1967, 1968; STRAUCH and PAGE, 1968; MACPHERSON and TIMUSK, 1970a). Changes of short-range and electrical polarizability have been seen to have little effect on the local-mode frequency in regions with strong nearest-neighbor force-constant changes, as needed for a reasonable description of the sideband spectrum. Moreover, they do not have any effect on the sideband spectrum itself, since the defect is at rest in evenparity modes. The monopole (breathing) deformation of the defect, though, if changed, changes the A1g (r;+) (breathing) vibrations, too, thus allowing a perfect description of the total spectrum with essentially one anharmonic and three harmonic coupling constants, Fig. 30.13e, see also below.

Comparison of the calculated sideband curves with the experimental ones shows that the latter usually show non-negligible absorption in the band-gap frequency region. This is due to coupling of more than one phonon to the local mode.

One also finds that peaks on the high-energy side of the spectrum have a slightly larger separation from the main line than those on the low-energy side. This is due to the frequency dependence of the local-mode shift function, which is larger at higher frequencies. Both these effects can be explained by higherorder approximations for the self-energy. In Sect. 30k it is indicated how these higher-order effects can be extracted from the experimental curve. The result is curve f in Fig. 30.13. It is this curve to which the calculations according to the simple equation above have to be compared.

k) Higher-order effects. The foregoing discussion 'of width and shift was based on the consideration of a few contributions to the self-energy. The discussion of the sidebands was even based on an approximate form of the anharmonic self-energy, although the form (30.52) presented some generalization to the inclusion of the off-diagonal terms in the self-energy. Processes E, F and G shall now be investigated.

In the same way as processes A and B are of the same order of perturbation theory and are competing processes, so do processes E and G compete with process D. Processes E and G and the interference process F have the same temperature dependence as D. Their contributions to the width and shift are quite analogous to those from process D, cf. (37.45)ff.

Taking account of these processes does not change the conclusions as far as sign or isotope effects are concerned. However, absolute magnitudes may be changed.

Another way of approaching higher-order effects is by inserting the anharmonic Green function into (30.20). The higher-order terms describe decay into many phonons by means of successive two-phonon decay processes. These processes have been found by TIMMESFELD (1968) to be important for the width and shift of low-frequency resonances. If process E shall be called the first iteration of process C, then TrMMESFELD's calculations converge after two to four iterations. This corresponds to four- to six-phonon processes. TrMMESFELD'S results for process C and its iterations are shown in Fig. 30.14. Actually the effect of process B is included in the curves of Fig. 30.14 even though the dependence

Sect. 30

1 IE

u

~

-0 .~

=a :I:

10

3

Anharmonic effects in perturbed crystals

-- Exp.curve ........... Not iterated - - Once iterated

I ----- Four times iterated 1/11

/;'~I 1/ I ... // III .

y~1 ......... . 1/ ....... /

I ... . 1/ /'

'l. ."" U···""

........................... -::;/ ... 0.3 L-____ ~----~=------l

2 50 Temperature(K)-

443

Fig.30.14. Higher-order effects in the half-width of NaCl:Cu+. In zeroth order the width results from process A (Fig 30.1). The first iteration gives a width resulting from processes A and E, etc.

(DMMESFELD, 1968)

on that process is only implicit. Since there is a sign error for process B, the effect of process A is simulated. In order to account for the correct shift, the contribution from A is then minus twice the contribution from process B.

The sideband spectra of the U center also exhibit features which indicate higher-order processes. In first approximation for I;.R(W) in (30.51) one would include only cubic anharmonicity, which leads in the sideband frequency region to absorption by the local-mode phonon plus one band-mode phonon. There is a frequency gap for the phonons in crystals whose atoms in the elementary cells are very different in mass, like KBr and KI. Indeed, this gap is seen in the Ucenter sideband spectrum. However, a residual absorption remains in the gap, indicating absorption by the local mode plus two or more phonons, cf. Fig. 30.13 a.

Sidebands appear as Stokes and anti-Stokes bands on both the high- and low-frequency sides of the local mode. They arise from excitation and annihilation of single band phonons, if one neglects the higher-order effects described above. The Stokes and anti-Stokes bands differ only by thermal occupation number factors in that approximation. Experimentally, peaks in the highfrequency sidebands frequently appear at slightly larger frequency separations (w+) from the main line than do the corresponding peaks in the low-frequency sidebands (with separation w_). This is because of the frequency dependence of the shift function LlRR(W). The argument is as follows: in the lowest order, the energy conservation in absorption processes implies a delta function with argument

in the damping function I;.R(W), (30.22).


The energy absorbed, h w, has to be equal to the energy of the local mode, hwR , with the energy of the band phonons, hwB , added or subtracted for the high- and low-frequency sidebands, respectively. In this approximation one has, in fact, w+ =W_.

To higher order, the argument of the delta function has to be replaced by

wR(W)±WB -w =WR + 2 WR LlRR(W)±WB-w

=2 wR LlRR(WR± w±) - 2 WR LlRR(WR) ± w B += w±.

In the frequency region of the high-frequency sideband of the resonance, the shift function is greater than in the low-frequency sideband,

LlRR(WR +W+»LlRR(WR -w_),

see Fig. 30.2b for an example. The peak separations from the main line are

w+ =WB +2 wR LlRR(WR +w+)- 2 wR LlRR(WR)

w_ =wB - 2 wR LlRR(WR -w_) + 2 wR LlRR(WR).

Since now from Fig.30.2b

LlRR(WR + W +) - LlRR(WR) > LlRR(WR) - LlRR(WR - W _),

one finds w+ >W_; the separation of a peak in the high-frequency sideband from the main line is larger than the separation of the corresponding peak in the low-frequency sideband.

Actually, there is not a delta function but a Lorentzian. As a consequence, the peaks in the sideband spectrum are "washed out". With increasing temperature not only do the (side)band phonons become broad, but the local-mode peak becomes decreasingly well defined. The width of the local-mode peak is thus

"20.8 "§ .ci

< .§ 0.6 a.

10.4

-·Kexp

--- K(O) x 0.59

600

Fig. 30.15. Experimental ("anharmonic") and "harmonic" sideband spectra in KBr:H-. See text. (ZEYHER and BILZ, 1969)

Sect. 31 Phonon frequency shift from bulk and local strain 445

reflected in the sideband spectrum. A very nice experiment of this kind was performed by HARRINGTON and WALKER (1970).

BILZ et al. (1967) and ZEYHER and BILZ (1968, 1969) proceed along the lines opposite to TIMMESFELD. On the assumption that the experimentally observed sidebands are caused only by the local-mode anharmonicity and not by that of the band modes, they calculate the pure absorption as it would appear if only the local-mode anharmonicity to lowest order were responsible. An example is shown in Fig. 30.15. There are remarkable effects: 1) The absorption in the band gap is strongly suppressed. 2) The separation of the low- and highfrequency sideband peaks from the main line is assimilated. In KCI the effect of the shift function is to shift the main sideband peaks from the "pure" positions w+ =56cm- 1, w_ =58cm- 1 to the experimentally observed values of w+ =64cm-t, w_ =58cm- 1 . It is the "pure" rather than the experimental curve to which any low-order perturbation calculation of the sideband shape has to be compared. Of course, a procedure along the lines of TIMMESFELD is far more desirable.

In the framework of the shell model, the polarizability derivatives, which govern the Raman scattering intensity, can be described by anharmonicities (Sect. 27 f). Therefore, multiple-order Raman scattering would, in principle, be a higher-order anharmonic effect. MARTIN (1976) has observed multiple scattering by a localized mode up to seventeenth order; the experiment is carried out close to resonance, in which case the model would be overstressed.

31. Phonon frequency shift from bulk and local strain due to temperature variation, pressure, and lattice distortion in defective crystals

a) Equilibrium positions. A phonon frequency is determined by the harmonic force constants and the phonon self-energy which contains the anharmonic force constants. The temperature dependence of the phonon frequency is determined by that of the self-energy and allows conclusions about the anharmonic force constants.

A second source of information about anharmonic parameters is obtained from the application of static external fields because the harmonic force constants as expansion coefficients of the total (lattice plus external) potential are functions of these fields. Actually, the force constants are functions of atomic positions which in turn are functions of the external fields. Because of the practical importance of pressure experiments for the determination of anharmonic force constants, this section will be devoted to the study of the atomic positions as a function of static external pressure and of their influence on resonance-mode frequency shifts. By use of the same formalism one can include the effect of temperature via thermal vibrations and of defect-induced lattice distortions. Since, among other approximations, sublattice shifts will be excluded, the effect of external electric fields will not be considered here. These effects should readily be accounted for by generalizing the theory below.

The calculation of the atomic positions is treated in Sect. 36b. For the calculation of lowest-order effects, it is sufficient to include processes A and B of Fig.36.2 contributing to the self-energy, and processes a to d of Fig. 36.1


contributing to the average displacement or deformation. All these processes involve potential derivatives of at most third order.

The qualitative aspects of the behaviour of the crystal are most easily explained following the formal treatment of Sect. 36. If the lattice potential is denoted by tP, the Hamiltonian of the lattice is T + tP and that of the lattice under the influence of external forces is

H = T+ tP- Ij(/) u(/) I

=T+tP-ju=T+W

with u(I)=u(L, IX) being the displacement of an atom at site x(L) in the IX

direction (in real-space representation) and j(L) being the external force (due to pressure) acting on the corresponding atom. The total potential is then W In order to reproduce similarity with Sect. 36 the harmonic Hamiltonian of the perturbed lattice is written as

H=H(O)+ V,

cf. (36.2)ff., with

where cpiO) here denotes the force-constant matrix of the pure lattice, denoted earlier in this chapter by CPo. The perturbing potential then is (neglecting higherorder terms)

where Vz contains the changes in the harmonic force constants introduced by the defects,

Of course, Vz could have been added to cpiO), see also below. V3 contains the (perturbed) anharmonic force constants,

Finally, V3=CP3= W3·

(31.1)

where cp 1 contains the internal forces which push the atoms into their equilibrium positions. In particular, these forces cause a lattice distortion around a defect as explained in Sect. 22 b.

One now writes the anharmonic Green function, GA, in the same spirit as in Sect. 36 and as a variant to the first sections of this chapter as

GA=[(GH)-l +1']-1 = [(CP~?)_Wz M)+ Vz +1'B]-l

corresponding to Fig. 36.2 or to (36.44). The term Vz can be correlated with the first contribution (diagram A of Fig. 30.2) to the self energy, L; the true anharmonic part is LB, the second contribution to L. LB has the form

to lowest order in V3 , or (31.2)


in real space representation, where (u(l) = (u(L, IX) is the average displacement of an atom. In the following one will have to calculate (u).

With the external perturbation j included in the potential W, (31.1), the average atomic displacements are obtained from (36.24),

(31.3)

One obtains (u) = - [ePiO)]-l [Vi + Vz (u) +t V3 (u u)]

in analogy to (36.28) or Fig. 36.1. The last term is represented by both diagrams c and d of Fig. 36.1; see also below. Alternatively one can add LlePz to ePiC) and obtains

Here [ePiC)] -1 =Go(W =O)=G~T

is the static Green function of the pure lattice, while

[WZ]-l = [ePiO) + LlePz] -1 = [1 +G~T LleP Z]-l G~T =GST

is that of the perturbed one.

(31.4)

The expansion of the potential W in terms of u so far is about an arbitrary set of atomic positions, x. A displacement, u, of an atom or ion now is a superposition of its vibrational displacement, v, and a static displacement, s, from the arbitrary center of expansion, x, to its actual equilibrium position, i.e.,

u=v+s.

The true equilibrium positions are then x + s. If the expansion were about the actual (pressure- and temperature-dependent) positions, one would have s = 0 and (u) =(v)=O.

This section will focus on the defect-induced distortion of the lattice as well as on the deformation of the perturbed lattice as a function of temperature and pressure. As the initial set of atomic positions, (x), one takes, therefore, those of the harmonic perfect lattice, which satisfy

(u)= -G~TW?).

These equilibrium positions cannot be defined uniquely, however, because WiD) vanishes by symmetry in an infinite lattice in which each atom is at a center of inversion. The equilibrium positions in a real crystal are determined by boundary conditions; these conditions may be that given external forces act on the surface or given surface atom positions. One need not specify the boundary conditions here, and so one keeps (x) as arbitrary. If, for example, (x) referred to the perfect (anharmonic) crystal at finite or zero temperature, then Wi would refer to defect-induced forces. In the following the expansion in terms of v + s is assumed to be about given equilibrium positions of the hamonic perfect crystal without external forces. The displacement field s then describes the static displacements of the actual temperature-dependent equilibrium positions from those of the harmonic perfect crystal. For the reasons mentioned above, the


average vibrational displacement <v) from these displaced posItIOns is then zero, whence one obtains with «v+s)2)=<vv)+2<v)s+ss=<vv)+ss, cf. (35.40),

Since one is interested in first-order effects one neglects the second term (corresponding to diagram c of Fig. 36.1) and obtains for s an expression which again has the structure of (31.4),

(31.5)

Neglecting higher-order terms seems to be an excellent approximation e.g. for the NaCl:Cu+ system (SMITH and SIEVERS, 1975), for the Mn defect in CdS and ZnS (ZIG ONE et aI., 1976), and probably for other cases, among them the defect halide ions in alkali halides, while anharmonic effects seem to play an important role for the F center (MORALES, 1981).

The deformation is now seen to have essentially two contributions. The first, from Wi' is independent of temperature and gives the defomation as a result of the external forces (j) due to pressure and of the internal ones (~i) due to the defects, see (31.1). The second term is temperature-dependent and describes the deformation of the crystal due to "thermal" forces originating from thermal fluctuations: with increasing temperature the atoms vibrate with increasing amplitudes. Pictorically, the atoms need more space in order to be able to do this. They repel one another by what one may call thermal forces. The effect is a (possibly non-isotropic) macroscopic expansion of the crystal with increasing temperature. Even at zero temperature there is a contribution to s from the zero-point vibrations.

In ideal crystals, in which one has ~ 1 = 0, the effect of the external and thermal forces is a uniform macroscopic deformation of the lattice given by

s(L, IX) = I eaf3 x(L, 13) f3

(31.6)

where x(L,f3) is as usual the 13 component of the (harmonic) lattice-site vector of the particle L, and where eaf3 are the components of the displacement gradient tensor, eaf3 = os(L, IX)/ox(L, 13). In addition, the sublattices of crystals of sufficiently low symmetry (e.g. piezo-electric crystals) may shift relative to one another. Inclusion of static external electric fields would also lead to a relative sublattice shift and, for piezoelectric crystals, to a uniform deformation. In order to keep the theory simple, relative sublattice shifts will be excluded, i.e. the effect of external· electric fields on crystals with certain low symmetries will not be considered. Also excluded will be the effect of inhomogeneous force fields which result in bending of crystals, for example.

As lang as one is interested in lowest-order effects, one employs a series of approximations which are explained in the course of this section. In particular, the finite strains will be approximated by infinitesimal strains. The infinitesimal strain eij=!(eij+e) is the symmetrical part of the displacement gradient tensor e; the antisymmetrical part of e describes rotations. Finite strains lJij are


defined as

and are symmetrical. Solved for the (pure) strains eij ,

and this inserted into (31.3) means that in the expansion of W in terms of 1Jij the expansion coefficients contain w" of different orders n. To first order finite and infinitesimal strains are equal, rotations being excluded from now on.

Equation (31.4) contains the anharmonicity only in lowest-order perturbation theory and only as regards the effect of temperature. In general, the elastic constants depend on temperature and stress (as a consequence of using finite strains instead of the infinitesimal ones) and differ for adiabatic and isothermal processes. However, these dependencies contribute to the phonon frequency shift only to higher order in the external pressure and anharmonicity. The infinitesimal strains are therefore sufficient as long as one considers phonon frequency changes to first order in the stress or temperature. In this respect this treatment is similar to that of MARADUDIN (1962) for the case of perfect crystals. The scope of the present article does not warrant to go into further details at any depth. For these points reference is made to LEIBFRIED and LUDWIG (1960, 1961); WALLACE (1965, 1970, 1972); BARRON and MUNN (1970); and BARRON et al. (1971). It seems that non-linear terms in the pressure have to be included for those defects which have a tendency to go off-center (KAHAN et ai., 1976).

Expressing, in pure crystals, the uniform deformation in terms of the (pure) strain tensor, e, and the uniform external and thermal forces in terms of the stress tensor, (1, rather than forces j one has

(31.7)

(31.8)

where Co and So are the tensors of the elastic stiffnesses and elastic compliances, respectively, of the perfect lattice. These tensors actually relate finite strains and the conjugate stresses. British and American usage differs here:

American British German

s

elastic compliance constant elastic modulus Elastizitatskoeffizient

c

elastic stiffness constant elastic constant Elastiziilitsmodul

Even though the strain resulting from the stress in a perfect crystal may not always be isotropic (because of non-equivalent crystallographic axes, or in the case of relative sublattice shifts), it is usually homogeneous. Introduction of impurities alters the situation in various ways: Due to the defect potential there are local non-uniform variations of the quantities cP 1 and cP 3 ; also the cor-


relations <v v) will show local variations. These effects combine to give a static deformation of the perturbed lattice in which the uniform (bulk) deformation is changed from that of the perfect crystal and is superposed by a local nonuniform distortion,

S = Sbulk + Sloeal = Stotal, (31.9)

Sbulk = ex. (31.10)

The strain, i.e. the homogeneous part of the deformation, will be shown in Sects. 31 c and e to be given by a relation like (31.8) with altered (bulk) compliance constants. The local distortions sloeal correspondingly will have to be expressed in terms of what are called local compliance (or elastic) constants. As a matter of fact, one of the problems to be solved is the separation of the homogeneous (ex) and inhomogeneous contributions (Sloeal) to the deformation; Sbulk is distinguished from sloeal by the fact that the former does not change from one lattice cell to another, while the latter has to go to zero for large distances from a defect. Each of the three terms in s,

S = -GST [4>l - j +t4>3<VV) ] (31.11)

is expected to contribute to both. Before the details of the effects of the internal forces 4> 1 (Sect. 31 d), of

pressure (Sect. 31 e), and temperature (Sect. 311) are investigated, the effects of defect concentration in the low-concentration limit need investigation (Sect. 31 b). In Sect. 31 c it will be attempted to establish a relation between the elasticity theory - with the result as in (31.8) - and the lattice theory - with the corresponding result as in (31.5).

Thermal expansion of a crystal with defects has been considered by ESHELBY (1956) using a continuum mode. IOSILEVSKII (1967) gave a description of thermal expansion of crystals with isotopic defects. SUBASHIEV (1969) considered the change of a resonance frequency as a function of both temperature and pressure. TIMMESFELD and ELLIOTT (1970) considered the temperature-dependent change in the lattice constant due to a defect to be based on two grounds: the local strain is changed due to 1) the altered anharmonicity 4>3' and 2) the altered harmonic force constants around the defect which enter into GST and <vv) of (31.5). Local strain due to altered zero-point vibrations of defects in quantum crystals has been considered by KLEMENS and MARADUDIN (1961); VARMA (1969,1971); and NELSON and HARTMANN (1972).

Pressure experiments have been analysed very often along the lines of the paper by GEBHARDT and MAIER (1965); see also KAPLYANSKII (1964); SCHNATTERLY (1965); NEWMAN (1969). There the response of the perturbed crystal to the external pressure has been assumed to be that of the perfect crystal. Although one knows that this may be a crude approximation, lack of sufficient information about the perturbations due to the defect has long made a more exact analysis impossible. BENEDEK and NARDELLI (1968a) have treated the bulk and local elastic constants of a perturbed crystal. Their results for the elastic response are in agreement with those of TIMMESFELD and ELLIOTT (1970). Further investigations have been carried out by LUDWIG (1968); PISTO-


RIUS (1970); LUDWIG and PISTORIUS (1972); SIEMS (1968); EISENRIEGLER (1971); and DEDERICHS and ZELLER (1972). The results of SUBASHIEV (1969), who also took into account the local alteration of the elastic constants, differ somewhat from those of the above authors as well from as our presentation. The effects of pressure and electric field have also been studied by MARADUDIN et al. (1967).

Electric-field effects have been investigated mainly for electronic transitions and for paraelectric impurities (for the latter see the review by BRIDGES, 1975). Electric-field-induced frequency shifts of localized modes were first observed by HAYES et al. (1965b) and HAYES and MACDONALD (1967). The Stark effect of resonant modes has been observed (KIRBY and SIEVERS, 1968, 1970; KIRBY, 1971 a; CLAYMAN and SIEVERS, 1968b; CLAYMAN et aI., 1971). The electricfield-induced absorption of light by even-parity resonances (forbidden without field) has been observed by KIRBY (1971 b), see also GANES AN (1971) for induced Raman scattering by odd-parity local modes in alkali halides. Electric field effects are becoming increasingly important as a means of investigating anharmonic properties; however, this article will be limited to the effects of external pressure and temperature.

b) Low-concentration approximation. In this section the static distortion of an impure crystal with a small concentration of defects is considered. A type of a coherent potential approximation for the elastic constants has been presented by DEDERICHS and ZELLER (1972; further references therein). Being interested in bulk and local strains, a different and in fact very simple-minded approach to the problem is taken here by considering the configurational average <s)av' The method to be used here is that of Sect. 28 a, and its application to strains shall be introduced by considering first the deformation

s(1) = - GST <P1

due to the internal force as an example. The suffix ST from the static Green function will be dropped. One wants to calculate the distortion near a defect at site x(~) from the forces <P1 due to a defect at site x(O. On configurational average one has

<s(1)~)av= - <2: G~?: <P1)av ?:

= -G~~<Pl

-p I G~L <Pf, L

(31.12)

where one now assumes a defect to be situated on every sublattice site with probability p. A factor of (1- p) has been omitted from the first term in accordance with the rules established in Sect. 28b. Let CP1 denote the non-zero submatrix of <Pf due to a defect at site x(L),

If L now denotes any arbitrary site, the vector <Pf does not depend on L and can be taken out of the averaging procedure.


The zero-frequency phonon is produced by the perturbation cPt at site x(L) and propagates to site x(~) where it causes a strain. On its way, it may be scattered at site x(L), then it propagates through the crystal, where it may be scattered off all the other perturbations, and finally reaches the site x(~) where it is scattered again. The result is the total deformation around a defect site, since no measures have been taken to subtract the homogeneous part. In mathematical terms, (31.12) then becomes

S~~lal=<s(l)~>av= -G~~ cPI-P L [1-G~~ T~] G~L[I- TLG~L] cPt. (31.13a) L

One uses the representation (25.9) of the unperturbed Green function; with the fact that (1- TL G~L) cPt is the same for each L and with (25.11) the configurational average results in

(31.13b)

In this equation one has restricted oneself to the calculation of the distortion in the impurity space.

The uniform deformation arises at any arbitrary site which does not belong to the immediate neighborhood of an impurity. Hence the zero-frequency phonon propagates as above but without final scattering. Following arguments analogous to those above, one obtains

(31.14)

That this part of the deformation is, in fact, of the form s = e x will be shown explicitly in Sect. 31c. In short, the reason is that @0(0) is independent of the cell index, while g and go decrease with increasing distance from the corresponding defect site. S~~lal as in expression (31.13) thus is in fact the total deformation, as expected.

Next thermal expansion will be considered. There are contributions to the deformation from the unperturbed anharmonicity, cP 3 0' at every lattice site and from anharmonicity perturbations, A cP 3 , in defect regions (not perturbed anharmonicities; this is similar to the reasoning connected with the absorption caused by charge perturbations in Sects. 25b and 28e). Arguments as above lead to the following results,

s~~lal = <S(3)~) av = - < L {I G~L cP~~~L" + L G~~ A cPSf' L"

L', L" L ~

+ G~~ A cP~L' L"} t< v(£) v(£') >av (31.15 a)

= -(1- got)@0(0)cP 3 , 0 t <vv>av

- p(l- go t) @o(O)(I-tgo) A 1P3 t<VV>I- g A 1P3 t< VV)I (31.15b)

with

being the change of anharmonicity in the impurity space which has to be chosen sufficiently large to contain A (j)3' t, and (j)1' Equation (30.24) can be used here


with the results

1 00

!< V (I.:) V(I.:')! = h - J dw(1 + nro)Im g(I.:, I.:'), (31.16a) n -00

1 00 1 _ !<v(I.:) v(I.:')av=h- J dw(1 +nw)Im N L ~o(q) expiq· [x(I.:)-x(I.:')].(31.16b)

n -00 q

Generally perturbations may occur in the long-range anharmonic interaction between a defect and the lattice, or between two different defects. For simplicity, only local anharmonicity perturbations have been included; the correlation between defects, to which the different arguments of tP3 may refer, has been neglected here.

The bulk deformation (expansion) is easily seen to be given by

S~~!k = -~0(0)[tP3 0 !<vv)av + p(l- tgo) L1 ffJ3 !<vv)a. (31.17)

For pressure-induced strain, finally, one obtains

S(2)=Gj=(Go-Go TGo}j=(I-Go T) Goj·

In an unperturbed crystal the deformation is

Sb2 )=Goj

so that the perturbed deformation can be expressed in terms of the unperturbed one (ELLIOTT et aI., 1968),

S(2) =(1- Go T) Sb2) =(1- GV) Sb2).

One obtains for the average distortion

S~;{a! = Sb2) - G~~ V~ s~) - p L G~L VL Sb2)

L

= Sb2) - gvsb2) - p(l- got) ~o(O)(I-tgo) VSb2)

=(1- go t)(I- p~o(O) t) s~)

and for the bulk deformation

S~~!k = (1-P ~ 0(0) t) s~).

(31.18 a)

(31.18b)

(31.18 c)

c) Relation between lattice and elasticity theory. The hypothetical infinite crystal with periodic boundary conditions cannot display the effects of thermal expansion merely because of its boundary conditions. The expansion of a real crystal can be interpreted as the excitation of some of its low-frequency acoustical modes (see also Sect. 36b). The frequency and displacement pattern depend on the shape and size of the crystal. Within a crystal volume element which has dimensions large compared with the effective range of interaction, i.e. with the atomic scale, but small compared with the crystal dimensions, the displacement pattern of any of these lowest-frequency modes is that of an acoustical mode with very long wavelength. In addition, except for volume elements very close to the crystal surfaces, this long-wavelength mode will look almost the same in every volume element at any arbitrary place in the crystal.


Besides the approximations mentioned earlier, surface effects will be neglected. As a matter of fact, the crystal properties in the interior of the crystal are essentially those of an infinite one. On the other hand, if a finite crystal is cut out of an infinite one extra stresses will generally be necessary to keep the atomic configuration of the finite lattice the same as those in the infinite one. Free surfaces then result in an extra (bulk) strain, the so-called image term (ESHELBY, 1956; LUDWIG, 1967). At the same time, the defect-induced strain has to be calculated using the appropriate Green function of the finite crystal (DEDERICHS and ZELLER, 1972). The result will be different from the strain in an infinite crystal. This difference partially cancels the "image" strain. Otherwise, it does not seem quite clear how important these two effects are. Surprisingly, the result (31.36) below (with Ji1- =1=0, Jill =0) for the relative volume change, (31.32), disregarding all surface effects, agrees with LUDWIG's (1967) result which explicitly includes the image term.

The idea of this section is to approximate the low-frequency behaviour of a real crystal by the long-wavelength acoustical-mode behaviour of a crystal element of an infinite crystal. (The atomic displacement field within this element is then ex as desired, relative sublattice shifts being excluded.) The results of this section will be somewhat sketchy but may, nevertheless, indicate how the Green function (rather than the dynamical matrix) might be applied to the theory of elasticity.

In the elasticity theory of unperturbed crystals strain and stress are related to one another by (31.7), (31.8),

Rather similarly to the perturbed case, (1 here contains two terms, one due to external pressure, i.e. external forces j,

(31.17 a)

(cf. (2.9) of LANDAU and LIFSHITZ, 1970), the other due to anharmonic effects, i.e. "thermal" forcesjA (MARADUDlN, 1962; for example),

(3)_ 1 " . (J~f3 -- L... x(L, a)1A(L, /3), V L

(31.17b)

j A (L, /3) = - I tP 3. o(L, /3; £, a' ; £1, cl')! < v(£, a') V(£', a"». (31.18) L'L"~'~"

V is the volume of the harmonic crystal without pressure. If there is no net torque on the crystal, one has

(J~f3 = (J f3~

in both cases.


In lattice theory the corresponding equations are from (31.3) and (31.5)

L qJo(L, /3; E, y) eoy~x(E, b)=j(L,/3)+h(L, p), L'y/j

s(L, oc)= L eol1.p x(L, /3) P

= L Go(L, oc; E, y)[j(E, y)+jA(E, y)]. L'y

(31.19)

(31.20)

If one multiplies (31.19) by x(L, oc)/V and sums over L, the result will be shown to be identical with (31.7). First one sets

! L x(L, oc) qJo(L, /3; E, y) x(E, b) = Co, I1.py~' V LL'

From its definition one can see that

Co, I1.Py~ = CO, ~YPI1.· From rotational invariance of the lattice it follows that

Co, I1.Py~ = Co, Pl1.y~ = Co, I1.P~y = CO, PI1.~Y·

(31.21)

to apparently has the symmetry of the elastic constants. The coefficients Co are defined in a way which implies a particular choice of the origin of the coordinate system. A representation of to independent of such a choice is as follows. First define

~ 1 ~ ~

Co, I1.py~=2(CO, I1.Py/j + Co, l1.~yp) 1

= --2 L [x(L, oc) -x(E, oc)J qJo(L, /3; E, y)[x(L, b)-x(E, b)J V LL' 1

= - 2 V/N t x(L, oc) qJo(L, /3; 0, y) x(L, b). (31.22)

Then define Co, I1.py~ = Co, I1.py~ + Co, p~yl1. - CO, PYI1./j·

From the properties of to it follows that

CO,l1.py~= CO,I1.Py/j·

The advantage of the coefficients Co rather than to is the independence of the former from the choice of the coordinate system.

In a crystal with periodic boundary conditions there cannot be any external forces. As for the thermal forces one writes Go using (25.9) as

G (1 I(; oc·[' 1(;' oc')=! "t§ (I(; oc· 1(;' oc'l q) eiq'[x(l,,,)-x(l',,,')) 0"", NL...o", , q

t§ 0(1(;, oc; 1(;', OC' 1 q) = (M" M",)-1/2 L e(I(;, oc 1 q,j) e*(I(;, oc 1 q,j) [w(q,j)] -2. j

(31.23)

As far as the different branches j are concerned, the three acoustic branches at q = 0 describe rigid lattice displacements, which do not lead to thermal expansion,


while the optical branches describe relative sublattice shifts (if any), again without expansion. As might be intuitively clear, the periodic boundary conditions prevent a consistent dynamical treatment of thermal expansion, because they lead to eigenfunctions with an even number of nodal planes which is a necessary consequence of the periodic boundary condition.

The argumentation so far holds for an infinitely large crystal. Surface effects as they occur for real crystals are going to modify matters. We shall not go into any details of boundary conditions but refer the reader to the above-mentioned literature, in particular to DEDERICHS and ZELLER (1972).

Consider now the solutions of the infinite crystal with periodic boundary conditions with periodicity length 10. The longest finite wavelength possible is about 10' the smallest wave vector component is equal to 2n/lo. The idea now is that the elastic properties in the interior of a very large crystal are essentially those of an infinitely large crystal. To put it in a different way, the nearlyuniform displacement pattern of a low-frequency eigenvibration of the infinite crystal near a node is extrapolated to large distances or alternatively to small values of Iql, much smaller than 2n/lo. In taking the limit of Iql tending to zero one certainly and on purpose violates the periodic boundary conditions, and one obtains the response due to a uniform expansion of the crystal with a hypothetical surface. Then for x(l, K) measured from the intersection of three nodal planes,

q . x(l, K) -+0.

The exponentials in (31.23) can be expanded, and from (31.20) one obtains

s(l, K, IX) = I Go(l, K, IX; 1', K', 1X')j(l', K', IX') l'K'a'

1 = N t I'~~Y +iq· x(l, K)+O(q2)} e(K, IX I q,j) M;1/2 w- 2(q,j)

. e*(K', IX' I q,j) M;, 1/2 {1- iq . x(l', K') + o (q2)} j(l', K', IX'). (31.24)

There are two different contributions to s. One is from the acoustical modes U = 1, 2, 3). In the limit of q tending to zero one has in crystals with high symmetry

e(K, IX I q,j)

VNMK

which is independent of K, with

(q-+O),

K

being the crystal mass and e(q,j) being the unit vector in the direction of e(K I q,j). In this case the summation over I' and K' can be carried out. Since there is no net force on the crystal, one has from the zeroth-order term

Ij(l', K', IX') =0 l're'

and for the first-order term with (31.17)

I qpX(I', K', f3)j(l', K', IX') = V I qp (JfJ~'. I'K'fJ B


As for the first pair of curly brackets, the zeroth-order term gives a contribution to s which is independent of I and K. This term represents rigid lattice translations which can be neglected 10. The first-order term results in s(L, a) proportional to x(L). Therefore, s is the displacement of an atom in a homogeneous deformation (strain) of the lattice. The strain is then proportional to the stress, and the proportionality constant is proportional to q q/w2 (q, j), which remains finite as q tends to zero. In the second contribution, from the optical modes 0=4,5, ... ), the quantity e(K, al q,j)/M;;1/2 does not depend on I either, but it does depend on K. The zeroth-order term in the first pair of curly brackets then describes relative sublattice displacements which will not be considered any further, in accordance with the restrictions listed earlier. This is where the theory is somewhat inconsistent. On the one hand we assume forces j acting on the crystal surface; on the other hand we take the Green function of the infinite crystal without surfaces and for wavevectors which actually do not occur.

With

and eap =Hos(L, a)/o x(L, [3) + os(L, [3)/0 x(L, a)]

one finds from (31.24) and by comparison with (31.7)

Here

1 - - - -So, apyil =-:j:{So. apyil + So, apily + So, payil + So, PailY)'

SO,apyil=lim ~ I:Ca(q,j) [Vp,j ~,j]-l eil(q,j). q~O P j

p=Mc/V

is the mass density of the crystal and

~,j=w(q,j)/qa (q-'O)

(31.25)

are the sound velocity components for the different acoustical branches. In this form the result can also be written as

- V 0 0 So,apYil=N ox(£,y) ox(L,[3) Go(L,a;£,blq-.O)

V 0 0 -,----- -,------,:- G (£' a' 0 b I q -.0)

- N ox(£', y) ox(£', [3) 0 ",

= - ~ ~ ox(1" y) oX(~" [3) Go(£', a; 0, b) (31.26)

10 That rigid crystal translations do not contribute to the shift of a phonon frequency is intuitively clear; it follows from the invariance of the force constants against rigid translations,

L <P3(L, IX; I'., IX'; I'.', IX") =0. L"

In fact, the rigid translations in the term

L <P 2 (L, IX; I'., IX') s(I'., IX') L'a'

in (31.3) give zero contributions for the same reason. They are thus to be excluded before (31.3) is solved for the strain to give (31.4).


with x(E')=x(L)-x(E)

(similarly DEDERICHS and LEIBFRIED, 1969). Note that So. aprfJ in the form of (31.25) depends on the direction along which q tends to zero. Perhaps one should take the average over all directions.

Inversely, one replaces q}0(0) as it appears in the results of Sect. 31 b by its long-wavelength limit and finds

<;§ o(K, oc; K', oc' I q =0) ~ lim <;§o(K, oc; K', oc'l q) exp iq . [x(L)- x(E)] q~O

= N I x(L, /3) So, app'a' x(E, /3'). V PP'

(31.27)

Being interested in pure strain and forces without net torque, one can replace So by So below. The representation (31.27) enables one now to substitute the elastic compliances wherever <jo(O) occurs in Sect. 31 b. As a matter of fact, one has to go back to the representations (31.13 a), (31.15a), and (31.18 a) instead of (31.13 b), etc. Go in the form of (31.27) still depends on the cell indices as in the representations (31.13a) etc., while in (31.13b) etc. the summation over the cell indices has been carried out. In particular one has from (28.7),

00=(1 +Go1')-l Go,

that the harmonic self-energy, 1', in the low-concentration single-site scattering approximation is given by a superposition of scattering matrices TL due to a single defect at all sites x(L) of the perturbed sublattice,

1'=p I TL. L

Because of the invariance against rigid translations one can write

1 N PVxITLX=p-xtx

L V

where on the right-hand side the vectors x have to be taken from the single defect site as the origin of the non uniformly expanding lattice. Except in connection with tP3 0 only the components of x in the impurity space are needed from now on: With <jo(O) from (28.19) and q}0(0) from (31.27) one finds

<j 0 (0) = [1 + pq}o (0) t] -1 q}o (0)

[ N ] -1 N = l+p V xSoxt V xSox

N [ N ]-1 =V x l+pSo V xtx Sox,

- N-q}0(0) = V xSox, (31.27')

in analogy to (31.27). If in the last expression for <jo(O), So is interpreted as the elastic compliance tensor of the defective crystal, then V (and x) refers to the


corresponding volume of the perturbed crystal as pointed out by LUDWIG

(1968). Writing Co = So lone has, in general,

Taking only N

p V xtx=LlC (31.28)

as the change in the elastic stiffnesses, one neglects contributions to Ll C of the order of (V - Vo)/Vo. LUDWIG (1968), defining C as the second derivative of the free energy with respect to strains, finds

[ N Ll V ] [ Ll V] - 1 LlC= p V XLllP2 X-J!; Co 1 +J!;

using i = x. This is different from our treatment, in which LllP2 would be replaced by t=LllP2(1+ gOLllP2)-1. This difference, as well as the use of i rather than x, is not entirely clear. For changes in the force constants appropriate to resonances of very low frequencies, Ll C in our approximation becomes large (and negative) due to the smallness of the denominator in t.

Now the local and bulk strains can conveniently be expressed in terms of the elastic compliances as

sl~lal= -glPl-p(l-got)xSo ~ x(l-tgo)lPl'

sl~lal=(l-got) (l-P ~ XSoxtfl xSotl

=(1- got) XSotl,

(3) 1 - 1 Stotal= -(I-got) V XSoX4»3,02(VV)av

N - 1 1 -p(l-got) V xSox(l-tgo) LllP32(VV)I-g Ll lP32(VV)I

=-gLllP3!(VV)I

-(l-got)xSO ~x[~ 4»3, o!(VV)av+p(l-tgo) LllP3!(VV)1

(31.29 a)

(31.29b)

(31.29 c)

Here one has used (23.4). The bulk strain tensors, by comparison with (~1.9) and (31.10), are seen to be given by

- N e(1)= -pSo V x(l-tgo)lPl' (31.30a)

(31.30b)

(31.30 c)


Writing So=So-SoL1CSo with L1C as defined in (31.28) and <vv)av=<vv)o +«vv)av-<vv)o) one finds

e = ell) + e(2) + e(3)

- - 1 1 =(l- SoL1C)eo-So V xcP3,OZ«vv)av-<vv)o)

- N 1 -pSo V x(1-tgO)(L1 lP3Z<VV)I+lPl)' (31.31)

The strain as expressed in (31.31) has four contributions which result from 1) the change in the elastic response (changed harmonic force constants), 2) the change in the harmonic dynamical properties of the crystal, 3) the change in the anharmonic force constants, and 4) from static distortioI'lS (change in first-order potential).

The volume change, incidentally, is

L1V= V· Tr(e-eo)'

The total strain, finally, can be written as

Stotal = (1- go t) Sbulk - g(L1 lP3 ! < vv) + lPl)

=(1- go t) x e - g(L1 lP3 !<vv) + lPl)'

(31.32)

(31.33)

The references of Sect. 29 c generally show the result that the shift of a mode caused by a given (average) lattice constant change is the same, irrespective of whether this change was achieved by external pressure or by adding further defects (in a mixed-crystal system). According to (31.33) this is indeed expected, since only the first term depends explicitly on both defect concentration and pressure, while the second does not depend on either. Likewise, if there are no changes in the anharmonic forces (L1</>3 =0), then a volume change can be induced by variation of either pressure or temperature (see TAYLOR et aI., 1975 for these effects on the impurity-induced spectra in KBr:TI+).

d) Static distortions. The strain due to static distortions is given by (31.29 a) and (31.30a),

To zeroth order in the concentration the strain around a defect agrees with that calculated in Sect. 22b. To higher order the strain from other defects is transmitted through the (perturbed) crystal and uniform strain is superposed. In the linear case the matrices t and thus g become scalars. As long as the impurity space is not distorted into a different symmetry (i.e. as long as det (1 + go v) does not have zeroes for negative frequencies because of too drastic changes v) the terms (l+gov) and (l+gov)-l go=g are positive. Since S is also positive, the local and bulk strain have the same sign, as is intuitively clear.

The lattice constant of mixed crystals varies with varying concentration of the constituents (GNAEDINGER, 1953; FUKAI, 1963a; PAUS and LOTY, 1965;


VON GUERARD et aI., 1969; for example). This has been used to observe the strain-induced shift of the U -center local-mode frequency (see references in Sect. 29c). The effect is twofold. Due to the change in the bulk lattice constants there is a uniform shift of the local-mode frequency with the mixing ratio. Secondly, because there are different numbers of one of the constituents on different neighboring sites of the U center, there are a number of additional local-mode lines.

Example: A mass defect with short-range force constants in a cubic surrounding

Here the example first used in Sect. 22 is taken up. The defect may have perturbed first- and second-order potential changes which are assumed to be restricted to occurring between the defect and its nearest neighbors. Denote by Afu and Ail- the change in longitudinal and transverse force constant evaluated at the undistorted lattice sites. Then the only non-zero components of CPt have also magnitude Ail-'

<)J1(100,x)= -<)J1(IOO,x)=roAil-' etc.; <)J1(100,y)=0, etc.

Group theory is useful in practical calculations. Odd-parity strains do not occur. Transforming to strains of a given representation, r, one has from Sect. 22d (dropping the suffixes local and (1) for the moment)

s(r, r, s) = L F(r, r, s 1 L, a) s(L, a) L, a

= L L F(r, r, s 1 L, a) [(1 + go v)-1]La, L'a' F(r, r, s' 1 £, a') s' LL'aa'

. L F(r, r, s' 1 M, [3) epp' x(M, [3'). MpP'

The last term can be written as

L F(r, r, s'l M, [3) epp' x(M, [3')= LXpp'(r, r, s') epp' MPP' PP'

where x(r) is a third-rank tensor in general. One takes [1 + go v]r and similarly g(r) and t(r) from (25.38). In the present model of the defect these quantities are scalars for the irreducible representations r/, r;i, rz~ of the group 0h' From Table 22.3 one also finds

and <)J1 (r) =0 for all other representations. For the x tensors one finds

x(r1 ) = ,~ ° 1 ° , 2 (1 ° 0) V 6 ° ° 1

(2 ° 0) x(r1i, 1)= ,rJ, ° -1 0,

v3 ° 0-1 (0 ° 0)

x(I;'i,z)=ro ° -1 ° , ° ° 1

(0 ° 0) x(rz~, 1)=rO ° ° 1 , 010

(0 ° 1) x(rz~, z)=ro ° ° ° ,

100


The change in the elastic constants then turns out to be

LI C"py~ = p N L x(L, a) t(L, P; E, y) x(E, c5) V LL'

N = p - L x(r) t(r) x(r),

V r

Sect. 31

These results are larger than those of BENEDEK and NARDELLI (1968a) by a factor of 2. The difference comes from an ad hoc assumption about the normalization volume VIN made by BENEDEK and NARDELLI. Experimentally determined elastic constants involve finite strains; see also the comments following (31.28). With

and

one has

SO(r1+)=SO,,,,,,,<x + 2So, <x"PP =Sl1 + 2s12 =(cll + 2C 12)- \

SO(r1~)=SO,,,,,,,,, -So,""PP=Sll -S12 =(C11 -C12)-l,

So(r2~)=4So,lZp"P=S44 =cii

(31.34)

(31.35)

(For the factor of 4 in SO(r2~) see, for example, NYE, 1969.) Finally, one obtains for the bulk strain tensor

( +_ (- - - ) 2rO 1 [1 (+) (.+)]-1 (+) e"p r 1 )- -P SO, "pxx+So, "pyy+So, "pzz V6 2r~ +go r 1 v r1 CP1 r1

= - c5"p L So(1~+) [1 + go(r1+) Llfll] -1 Llfl. = c5"p e(r/), ro

(31.36 a)


and for the local strain around an impurity

s(rt) = [1 + go (r1+) LlhJ -1 3 ~ e(rt)

- [1 + go (rl+) .dh] -1 go(r1+) V6 ro.dfJ..

,/C. [+ So(I~+)] =-v 6 rog(~ )+p[1+go(~+).dfll]2 .dfJ..· (31.36b)

e) Pressure-induced frequency shift

There are a number of experimental investigations of pressure-induced frequency shifts. The shift of local modes has been investigated by HAYES and MACDONALD (1967); HAYES et al. (1965b); FRITZ et al. (1968); BAUERLE and HUBNER (1970); and DOTSCH (1969a). Pressure experiments on resonant modes have been performed by NOLT and SIEVERS (1966, 1968); KAHAN and SIEVERS (1971); and KAHAN et al. (1976); see also BENEDEK and NARDELLI (1966a); BUSSE et al. (1968); BUSSE and HAIDER (1970); TIMMESFELD (1970); and GENZEL et al. (1969).

The strain due to external pressure is given by (31.29b) and (31.30b),

sl';lal =(1- go t) xe(2),

e(2)=SotJ.

In this section the suffices from sand e will be suppressed. Differently from S(1)

and S(3) the local shift S(2) is given entirely in terms of the strain, i.e. without additional contribution from local forces.

Example,' The frequency shift of an Einstein oscillator in an alkali halide lattice

The effects of defects on the pressure-induced strain are best visualized by applying a simple model system to a phonon frequency shift. Therefore, the pressure-induced shift of an infrared-active (threefold degenerate) resonance mode of a substitutional impurity in an alkali-halide lattice will be considered. Let the resonance be approximated by an Einstein oscillator resonance (i.e., by a resonance of very low or high frequency) in which only the defect takes part in the resonance vibration. Then the frequency of the resonance is obtained by diagonalizing the 3 x 3 matrix with elements

<1)2 (0,0:; 0,0:') + L <1)3(0,0:; 0, 0:'; L, {3) s(L, f3) =m(wiAxa' +.d w;a') (31.37) LP

where m is the defect mass and wR is the resonance frequency under zero pressure.

If one now assumes that the anharmonic interaction is mainly between the defect and its nearest neighbors, then the sum over L is restricted to these neighbors and only local effects are important. Even if this approximation is made, there are in general non-zero anharmonic coefficients <1)3(0,0:; 0, 0:'; L, {3) for 0: =l= 0:', which lead to a mixing of the three degenerate resonance vibrations.


Upon substitution for s from (31.29b) one obtains

milw;",= L L <1>3(0,0(; 0, 0('; L, f3) x(L, y) y~. Lp

. { (L1 ) L [(I+G~TVST)-lJL'P;L"P'X(E,y)So,p,y~.}(j~. x ,Y L'fJ'

= I L <1>3(0,0(; 0, 0('; L, f3) x(L, y) Spy~.(L) (j~ •. ~. LfJy

Sect. 31

The term in curly brackets can be interpreted as an element of a local elastic compliance tensor. This tensor as well as the anharmonic expansion term is perturbed by the presence of the defect. Alternatively one can group together tf>3 and (I + Go V)-l and interpret this as an effective anharmonicity. In any case, the formal structure is the same as that in the pure crystal.

The range of anharmonic interaction is now assumed to be restricted to the defect's nearest neighbors. The only non-zero force constants are then

a= <1>3(0, x; 0, x; 100; x), b = <1>3(0, x; 0, x; 010, y),

c = <1>3(0, x; 0, y; 100, y) (31.38)

as in Sect. 27f. It is also assumed that only the central and non-central force constants between the defect and its nearest neighbors (h and fJ.' respectively) are perturbed. Then the matrix algebra becomes very straightforward, and one obtains for crystals with the NaCI-structure

mil w;x = -tro(a + 2b) S(rl+)((j xx + (jyy + (j zz)

+-tr o(a - b) S(rii)(2(j xx - (j yy - (jzz)'

mil W;y =rOcS(r2~)((jxy + (jyx) (x* y) with

S(r)= So(r) 1 + v(r) go(r)

The bulk compressibility of the perturbed crystal, incidentally, is

K=3So(rl+)

(31.39 a)

(31.39 b)

(31.40)

(31.41)

but the important quantity here is the "local" compressibility, S(r/), which generally is a tensor.

Upon application of hydrostatic or uniaxial pressure the eigenvalues of the matrix (31.37) and thus the resonance-mode eigenfrequencies shift and split. For the various directions of the applied pressure p the matrix u with components

(1 0 0) U= -p 0 1 0

001 for hydrostatic pressure

(1 0 0) U= -p 0 0 0

000 for p II [100J


.~ -~(~ 1

~) 1

0

for p II [110]

.~ -} (: 1

:) 1 1

for p II [111].

After substitution of these expressions into (31.39a) and (31.39b) and of these two into (31.37) the eigenvalues and eigenvectors of the matrix (31.37) are easily found. The results for the frequency shift for pressure along different directions (p), and for various polarizations of the electric field vector (E) of the exciting light are summarized in Table 31.1. It is customary to express the frequency shift in terms of stress coupling coefficients A, B, and C, rather than of anharmonic coupling constants a, b, and c. The latter are related to one another in the present approximations by

h ro 2 A=- - - (a+2b),

2WR m 3

h ro 1 B=---(a-b)

2WR m 3 '

h ro C=--2c.

2WR m

Figure 31.1 shows the frequency shift of the local mode in KCI:H- versus pressure. Within experimental error the shift is linear in the pressure. There are four curves from which the three constants a, b, and c can be calculated. It turns out that the values obtained in this way are not uniquely defined. This is basically an experimental uncertainty since there are only three different experiments because of symmetry. In addition, the importance of the anharmonic interactions of the defect with neighbors other than the first as well as inclusion of the motion of the defect's neighbors in the resonance mode will

Table 31.1. Summary of frequency shifts of an Einstein oscillator in an NaCI crystal for pressure and polarization in various (parallel and orthogonal) directions. For notations see (31.38), (31.40),

and (31.35)

p E 3mdw~ ----

2 ro dp

Hydrostatic Any 3(a+2b) S(It)

[100] [100] (a+2b) S(r,+)+2(a-b) S(rii) [010] (a+2b) S(r,+)- (a-b) S(r,j,)

[110] [110] (a+2b) S(r,+)+4(a-b) s(r,~)+ic S(r;~) [110] (a+2b) S(r,+)+¥a-b) s(r,~)-ic S(r;~) [001] (a+2b) S(r,+)- (a-b) S(I-;~)

[111] [111] (a + 2b) S(r,+) + 2 c S(r;~) [110] (a + 2b) S(r,+) - c S(r;~)


0.8 KCl :H- . Ir 0.7 MAIN B~ND 502 cm-'I/~ ~IOO] 0.6

.. 0.5 E u

£0.4 :.<: en

03

0.2

T : 77 K E [100]

P[IOO] ECOIO]

10.0. 150. 20.0. 250. Stress, Kp/cm 2

Sect. 31

Fig_ 31.1. Frequency shift of the local mode in KCl: H - versus uniaxial stress. (FRITZ et a!., 1968)

influence the numerical results for a, b, and c. Also, the elastic constants may need correction further away from the defect. Internal electronic degrees of freedom of even parity ("breathing" etc.) influence the elastic properties of the host crystal and the defect region. This is a field for future study. Even though definite conclusions can, therefore, not be drawn, one can see that the coefficient a is larger than coefficients band c. This has implications, among others, on the relative importance of the different symmetries on the localmode sideband spectra (Sect. 30j), as these are determined by the same anharmonic coupling constants. Any further analysis of the effects of lattice dynamical models, local elastic constants, and the inclusion of anharmonic coupling to neighbors beyond the nearest ones requires more involved calculations.

f) Thermal expansion. Thermal strains are described by (31.17) and (31.18). The low-concentration result is from (31.29c) and (31.30c)

Sl;[al =(1- go t) xe(3)- g LllP3 !(VV)I'

(3) _ - N [1 1 1 ] e --So V x N 4)3,0 Z(VV)av+p(l-tgo) Ll lP3 Z(VV)I .

The contribution to the phonon frequency shifts is calculated from (31.37). If S(3)

from here is inserted into (31.37) the resulting expression has hardly any similarity with the corresponding contribution to the self-energy, second term of (30.20'), except possibly for the second term of Sl;[al. The main part of the static Green function has merged into the elastic compliance tensor.

Equation (31.30c) describes the bulk expansion as originating from the average anharmonicity (square brackets). The crystal reacts as an elastic medium with average compliance constants So.

The thermal expansion coefficient is given by

a In V a Tre !X=ay-=---ay-.

Sect. 32 Finite crystals 467

The only temperature-dependent quantity here is <vv) which becomes for high temperatures

cf. (30.49).

32. Finite crystals

a) Introduction. Several approaches have been used to calculate the modes of vibration of finite crystals. The continuum theory (FUCHS and KLIEWER, 1965; ENGLMAN and RUPPIN, 1966) treats the crystal as a homogeneous material with a frequency dependent, complex dielectric constant. The crystal is assumed to be capable of supporting a polarization wave, but its atomicity in other respects is ignored. Surface effects occur in this theory because of the proper application of boundary conditions. Since the continuum theory can be used to calculate only the long wavelength modes of macroscopic samples, this approach is particularly appropriate for the interpretation of the infrared absorption of finite crystals.

The second approach to the modes of vibration of finite crystals has been to calculate the eigenvectors and eigenvalues of the dynamical matrix of a lattice with a limited number of atoms in one or several dimensions (LUCAS, 1968). This approach has been used for samples of slab geometry containing up to 21 layers of atoms (CHEN et aI., 1971) and for small crystals containing up to 180 atoms (GENZEL and MARTIN, 1972b).

Although the validity of this approach is not limited to relatively long wavelengths or to samples of simple shape, experiments are normally performed on samples containing several orders of magnitude more atoms than can be handled presently in a computer calculation.

A third approach, using Green functions, can be used to calculate the modes of vibration of a semi-infinite crystal (LIFSHITZ and ROSENZWEIG, 1948). This is exactly the same approach as is described in Sect. 23. It has been shown in this section that the number of degrees of freedom in the final form of a defect problem can be reduced by using Green functions to the number of degrees of freedom needed to define the defect. If the defect is a surface, the number of degrees of freedom needed to define the defect is infinite. However, the translational symmetry of the surface in two dimensions allows the usual reduction of the problem into plane waves. Although this trick allows one to handle the two dimensional extension of the surface defect, in order for the problem to be tractable, changes in force constants far from the surface must be ignored. In spite of this deficiency BENEDEK (1976) has shown that the method can even be applied to ionic crystals. This is possible because the interplanar potential between atomic layers decreases experimentally with the layer separation. The formal elegance of the Green's function method makes it particularly appropriate for studying inelastic atom scattering from surfaces (BENEDEK, 1975).

b) Continuum theory of finite crystals. In the harmonic approximation a large ionic crystal absorbs light only at WTO' the frequency of the long-wavelength transverse optical mode. If the crystal is small, the frequency of the peak


absorption is increased by the polarization charge at the crystal surface. The designation large or small compares the size of the crystal to the wavelength of light in the surrounding medium. FROHLICH (1948) was the first to point out this effect. He showed that the mode of vibration with uniform polarization in a small ionic sphere has a frequency

(32.1)

The high- and low-frequency dielectric constants of the sphere have been denoted by eoo and eo. The dielectric constant of the surrounding medium is em.

FUCHS and KLIEWER (1968) and ENGLMAN and RUPPIN (1968) have extended this work by calculating the scattering and absorption by polar slabs and spheres of arbitrary size. Their calculations predict the existence of a series of surface modes with frequencies between Ws and WTQ. Ws is defined as the frequency where the real part of the dielectric function e(ws) equals - em. The Frohlich mode, although not localized at the surface, is the lowest mode of this series and is therefore often called a surface mode.

Since RUPPIN and ENGLMAN (1970) and KLIEWER and FUCHS (1974) have recently given complete reviews of the continuum theory of polaritons in finite crystals, the same material will not be covered here. However, it may be instructive to consider a limiting case (GENzEL and MARTIN, 1972) of this general theory, applicable to polar spheres much smaller than the wavelength of incident light.

Consider small absorbing spheres with a complex dielectric function e(w), embedded in a non absorbing medium with dielectric constant em. An average dielectric function eav for this composite medium may be defined. This dielectric function is defined in terms of the average electric field, Eav' and the average polarization field, p..v

(32.2)

It will be assumed that the average field is merely a volume average of the nearly constant field inside the spheres, E;, and the nearly constant electric field outside the spheres, Em' that is,

(32.3)

where f is the fraction of the total volume occupied by the spheres. Similarly, the average polarization field can be written,

(32.4)

The field E; inside a sphere in the presence of a constant far field Em is known to be

(32.5)


Using the last four equations, an expression for Bav is obtained:

(32.6)

This average dielectric constant determines the optical behaviour of small interacting particles. Clearly, Bav has a pole at the frequency Wr defined by the equation

(32.7)

Small crystals should have their maximum absorption at this frequency. For low-density samples, f approaches zero and Wr approaches WS' For high-density samples, f approaches unity and Wr approaches W TO '

. The infrared absorption of small particles was studied by MARTIN (1969, 1970) and LUXON et ai. (1969). Qualitative agreement with (32.1) was found; the peak absorption occured near W r ' and this frequency had the predicted dependence on Bm' However, the absorption band was much broader than predicted by theory, and there was an additional unexpected absorption peak at WTQ. The reason for these discrepancies is that the microcrystals are usually not spherical and they tend to clump together. These effects are illustrated in Fig. 32.1. The transmission spectrum of 1000 A cubes of LiF isolated in solid argon is shown in the upper curve. The isolated cubes have two absorption peaks at 450 and 550 cm -1, between WTQ and wLO ' Also shown in this figure is the transmission spectrum of LiF cubes prepared in the same way but not isolated in solid argon. This second sample consists of loosely packed clumps and chains of microcrystals. The aggregated cubes have a broad absorption which extends throughout the frequency region of negative dielectric function. This broadening is due to dipolar interaction between neighboring particles. CLIPPE et ai. (1976) have calculated the absorption spectrum for small spherical particles aggregated into various regular clusters, e.g., pairs, triplets, chains, etc. They show that simple chains have two resonances, one on each side of the resonance frequency of a single sphere.

The sample composed of isolated cubes is better defined. Therefore, its absorption spectrum can be calculated more precisely. Several papers have appeared recently reporting calculations of the infrared absorption spectra of microcrystals with shapes other than spherical (VAN GELDER et aI., 1972; LANGBEIN, 1976). In particular, FUCHS (1975) has developed a method to calculate the optical properties of particles with arbitrary shape. He first defines a particle susceptibility, X(w) in terms of the induced dipole moment, M, and the external electric field, EO,

(32.8)

where v is the volume of the particle. FUCHS then goes on to show that this susceptibility can be written in the form

(32.9)

1.5

1.0

0.5

300

200

100


The constants ni and ci depend only on the shape of the particle and have been calculated explicitly for the cube. The parameters ni determine the frequency of resonances in the susceptibility and the parameters Ci the strength of these resonances. It turns out that for a cube only six resonances contribute appreciable to the particle susceptibility. The position and the absorption strength of these resonances for the case of LiF are shown in the lower half of Fig. 32.1 along with the calculated total absorption coefficient.

When considering a slab geometry, it is not useful to take the simple electrostatic approach used above for small crystals. Because of the larger dimension of a semi-infinite slab, retardation effects must be included. That is, one should solve the equation of motion of the crystal together with Maxwell's equations. This can be done in a long-wavelength approximation. The resulting polariton modes can be classified as being either bulk (oscillating fields inside) or surface (decaying fields inside), and either radiative (oscillating fields outside) or nonradiative (decaying fields outside). Semi-infinite slabs have only nonradiative surface polariton modes. Excitation of these modes has been observed in electron scattering experiments (BOERSCH et aI., 1968; IBACH, 1970). Optical excitation of these modes is complicated by the fact that a plane wave will not couple efficiently to the decaying external field of the nonradiative mode. RUPPIN (1970) has suggested the technique of attenuated total reflection to

'-<l>

.Q

E ::3 Z <l> > 0 3:

Fig. 32.1

~ ~L I...--ko=w/c I [CfIT

400 _____ L _____

til, I I I I I I I

380 I I I I

tilT I I I I 360 I

1.0

Fig. 32.2

. .

-- Theory ... Experiment

1.5 2.0 Wave Vector k

Fig.32.1. Transmission spectrum of 100 A LiF cubes isolated in solid argon (upper heavy curve). Transmission spectrum of 1000A LiF cubes with no solid argon to separate the microcrystals (upper light curve). Calculated absorption coefficient (lower curve) of LiF cubes composed of contributions from the six surface modes (histogram) with the largest dipole moments. (MARTIN

and SCHABER, 1977)

Fig. 32.2. Experimental and theoretical dispersion of surface polaritons in GaP. Experimental accuracy is given by the size of the rectangles. (MARSCHALL and FISCHER, 1972)


overcome this. The evanescent wave of the external radiation couples efficiently to the evanescent wave of the surface polariton. FISCHER and MARSCHALL (1972) performed such an experiment on a GaP surface. The comparison between experiment and theory is shown in Fig. 32.2.

c) Lattice dynamics of finite crystals. An approach quite different from the continuum theory is to calculate the eigenvalues and eigenvectors of a crystal containing a limited number of atoms. The density of states and the infrared absorption calculated using a rigid-ion model are shown in Fig. 32.3 a. The rigidion parameters derived for an infinite RbF crystal have been used in this calculation to describe a microcrystal with extremely small dimensions, 6 atoms long, 6 atoms wide, and 4 atoms high. The density of states is found to resemble closely the density of states obtained using cyclic boundary conditions. The infrared absorption, however, is quite different. There exist three absorption bands: one at the long-wavelength transverse-optical mode frequency roTa and two more between roTa and roLO ' An examination of the eigenvectors of the modes responsible for this absorption shows that they can be characterized as either edge, surface or volume modes.

When the density of states of an infinite crystal is subtracted from the density of states of a crystal containing 180 atoms, a difference spectrum is obtained which more clearly shows the effect of surface (Fig. 32.4). RIEDER and HORL (1968) have measured the density of states of small MgO crystals using neutron scattering. The smallest crystals measured had an average dimension of about 100 A, i.e. 50 atoms on an edge. They subtracted the measured density of states of the bulk crystal from the density of states of microcrystals to obtain the

12 X 103 6 X 10-14 RbF .. /

wT wL

, 5 E -~10

OJ DO z LTIs !:!:! 8 !il 4 ~ < ! ~ 4 S

6 IJJ IJJ

U)

~ LL IJJ ~ 0

~ 4 (!) a: 3

~ 2 ~ ~ ~ ii: U)

i 2

ffil 0

2 3 4 2 4 5 x 10'3

FREQUENCY (rod/sec) FREQUENCY (rod/sec) Fig. 32.3. Calculated rigid ion density of states of a RbF microcrystal with atomic dimensions 4 x 6 x 6 (solid line) and RbF using cyclic boundary conditions (dashed line). Calculated infrared

absorption coefficient (lower curve). (MARTIN, 1973)


experimental difference spectrum shown in Fig. 32.4. The similarity of the shapes of the calculated and experimental curves is immediately apparent.

The calculation of the lattice dynamics of an ionic slab is complicated by the necessity of making two-dimensional dipole sums. However, this problem has been solved and extensive calculations have been made on several materials (TONG and MARADUDIN, 1969; CHEN et aI., 1970, 1971). One example, the dispersion curves of a 1S-layer slab of RbF, is shown in Fig. 32.5. Since the slab has translational symmetry in two directions, one must calculate the modes for a sampling of k-vectors in a two-dimensional Brillouin zone. The unit cell of such a slab contains 30 atoms, two atoms in each layer. Therefore, the dispersion

70

3 o 9 72 7S 18 frequency (10'+-odI5)--

Fig. 32.4. Calculated difference spectrum (solid line) obtained by substrating the rigid ion density of states of MgO with cyclic boundary conditions from the rigid ion density of states of a microcrystal (GENZEL and MARTIN, 1972b). The experimental difference spectrum was obtained from

RIEDER and HbRL (\968)

W (1013 rad sec-I) 6 ~~T>-rrT'-rT~-r~~.-.-.-~~-.-.-.-r'-'-T

40" -yy 20" 10"

5

4

2

l.=q OI2TT

Fig.32.S. Dispersion curves for a 15-layer slab of RbF with (100) surfaces, calculated with an 11-parameter shell model. Each pair of surface modes is labeled Si (i=I,2, ... ) according to an arbi-

trary scheme. (CHEN et aI., 1977)

Sect. 33 Present and future problems in lattices with defects 473

curves contain 90 branches. Several of these branches represent surface modes lying in the gap between bulk modes. The surface modes include the Rayleigh modes, S1' and the macroscopic surface modes, S 2' discussed in Sect. 32 b. In addition, the lattice dynamical calculations yield surface modes, S4 - S6 which are not predicted by continuum models.

33. Present and future problems in lattices with defects. To obtain an understanding of the processes in crystals which give rise to phonon spectra, models are apparently extremely useful. These models are, however, often difficult to justify physically and tend to become rather complicated. For example, one model may be needed to predict a full set of crystal normal modes, another to allow for the effect of a defect in the lattice giving rise to the observed spectra, and a third to give the coupling between the various excitations (phonons, photons, electronic excitations, etc). One important goal is to relate the various models to each other and, in particular, to basic physical quantities. Only when a model has such a good physical basis that many experimental effects can be calculated within its framework can it be really convincing, and lead to a better understanding on a microscopic basis of the processes involved in the interaction of light and solids.

An attempt has been made in this article to indicate the present level of work directed towards such an integrated model. The problems are, however, many: often models require parameters whose physical meaning is unclear; or, although the physical meaning seems clear on a simple picture, it is difficult to justify them quantum-mechanically (for example, the "breathing" deformability of negative polarizable ions). Also, many problems become unwieldy unless treated in a low-order approximation. Examples are: the problem of defects in a lattice, which may involve long-range changes (changes of charge from that of the host, large distortions of the lattice, etc.); the treatment of anharmonic effects; the treatment of heavily perturbed crystals, mixed crystals, and amorphous substances; and so on. This, of course, leads to problems in interpretation, and questions as to whether it is the model or the approximation that is inadequate.

One important aspect of the problem is to obtain a "microscopic theory". The quantum-mechanical derivation of force constants is a current problem. A more complicated one is the quantum-mechanical calculation of the changes in force constants introduced by a defect, and even more complicated is the quantum-mechanical foundation of the shell-model concept in pure and also perturbed crystals. A promising approach is that of Wood and coworkers (WOOD and OPIK, 1967; WOOD and GILBERT, 1967; WOOD, 1968; WOOD and OPIK, 1969; OPIK and WOOD, 1969; WOOD and GANGULY, 1973; MOSTOLLER and WOOD, 1973; see also MALKIN, 1969) to the problem of the U-center in alkali halides on a microscopic basis: they found that a proper orthogonalization of the H - wave functions with respect to those of the neighboring ions was necessary, and that the distortion around the defect led to an appreciable polarization of the U center.

In the dielectric formulation (see Sect. 6b) it is easy to see that defect-induced changes in the dielectric function lead to force-constant perturbations (FALTER and SELMKE, 1980; FALTER et al. 1981); but it is also clear that the

474 Dynamical theory of interacting phonon systems Sect. 34

analysis should be carried out in a real-space (rather than momentum-space) representation.

Further developments in the problem of the microscopic theory of the defects in lattices may be expected to be along such lines as those used by these workers.

G. Dynamical theory of interacting phonon systems

34. Basic concepts

a) Introductory comments. This chapter gives a condensed and rather selfcontained description of phonons and their interactions in terms of phonon propagators and functionals. The results of these formal treatments of selfenergies, Dyson equations etc. were used in the foregoing chapters, especially D and E, but the derivations are given in this chapter, where the full many-body theory is developed. This separation is convenient for readers who are primarily interested in the results and their applications to real crystals and who do not want to get involved in detailed mathematical analysis. On the other hand, an explicit discussion of the general equations of motion for phonon propagators and the techniques for solving them is of interest to readers who look for more general results and more extended applications of the theory.

The fundamental problem of lattice dynamics is to determine the phonon frequencies and the corresponding eigenvectors characterizing the displacement field of a mode. The interactions between the atoms of a crystal lattice, leading to the elementary phonon excitations as well as to phonon-phonon interactions, involve typical problems of many-body theory. The methods of calculation depend on the kind of lattice potential. For many problems of lattice dynamics, the lattice potential energy taken as harmonic in an initial approximation, and the anharmonic parts of the potential are treated by perturbation theory (BORN and HUANG, 1954; LEIBFRIED, 1955; LEIBFRIED and LUDWIG, 1961; COCHRAN and COWLEY, 1967; LUDWIG, 1967; COWLEY, 1968). Crystals with strong anharmonicity, large zero-point fluctuations, and unstable modes require nonperturbative and self-consistent methods (CHOQARD, 1967; WERTHAMER, 1969). A variety of special and general approaches, developed for this purpose, and an extensive body of literature can be found in the papers of NOSANOV (1966), GOTZE (1967), GOTZE and MICHEL (1968), HORNER (1967, 1970), KOEHLER (1968), PLAKlDA and SIKLOS (1969), MEISSNER (1970a, 1970b), KERR and SJOLANDER (1970), WERTHAMER (1970), TAKENO (1970), KOEHLER and WERTHAMER (1971), PYTTE and FEDER (1969), and FEDER and PYTTE (1970). In the following, we try to combine the perturbative and the self-consistent approaches. Most of the equations can easily be interpreted in either sense. The formal presentation is made in a manner such that the equations can be applied to perfect lattices as well as to lattices with defects. For various theoretical tools

Sect. 34 Basic concepts 475

which are not mentioned here, the reader is referred to the monographs and articles on many-body theory of KADANOFF and BAYM (1962), BONCH-BRUEVICH and TYABLIKOV (1962), ABRIKosOV et al. (1965), KWOK (1967) and FETTER and W ALECK A (1971).

The approach and the final approximative results of the theory imply the general assumption that phonons are well-defined elementary excitations with reasonably long lifetimes, even in the presence of anharmonic interactions. Effects of finite phonon lifetimes and of anomalous lineshapes are included by expressing most of the results in terms of the exact one-phonon spectral density function. Dynamical properties of a crystal, such as the excitation spectrum or transport coefficients, are calculated most directly from equations of motion in one or the other form. The Green function method, which is very convenient for solving equations of motion, will be discussed in detail in this chapter. Dynamical properties depend to some extent on static properties, which enter the equations of motion and the solutions as boundary conditions and parameters; for instance, the temperature-dependent structure of a crystal. In general, one has to treat these properties in agreement with thermodynamic requirements.

The dynamics of a lattice are here described by three sets of functions (which are not independent but related in various ways, such as by the equations of motion): (a) the time-ordered many-phonon Green functions, (b) a set of n-point functions following from a cluster expansion of the Green functions (sometimes called cumulants), and (c) a set ofrenormalized dynamical interaction coefficients (vertex functions). These three sets of functions are the functional derivatives of three different generating functionals, which are defined by introducing an external time-dependent source field and by generalizing three characteristic quantities used in statistical mechanics, namely, the partition function, the free enthalpy, and the free energy. General and exact relations between the different functional derivatives can be deduced in a straightforward way. Approximations are discussed in view of applications and for comparison with perturbation theory and diagram techniques (MARADUDIN and FEIN, 1962; CoWLEY, 1963).

In the treatment of absorption and scattering of external radiation by phonons in a dielectric crystal, one can distinguish two different regimes with respect to the phonon-phonon interactions. The first regime is defined by processes in which only a few phonons are involved; the amount of transferred energy is of the order of a typical phonon energy. These are the circumstances under which one observes the one-, two-, and three-phonon spectra in infra-red absorption, and Raman and neutron scattering. The phonon-phonon interactions, leading generally to frequency shifts, finite lifetimes, and lineshape anomalies, can normally be treated in this regime by low-order perturbation theory. An example for an exception is the effect of two-phonon bound states (RUVALDS and ZAWADOWSKI, 1970, 1971). If however, in the process of absorption or scattering, the transferred energy is much smaller than the average linewidth of the thermally excited phonons, the collisions between the phonons become very important. In this so-called collision-dominated or hydrodynamic regime, it is no longer possible to specify the final state of the crystal in terms of a few phonons, and perturbation theory fails completely; this does not mean that the phonons themselves are not well defined. Physical effects in this regime are the


dielectric losses of perfect crystals at microwave frequencies and the quasi-elastic Rayleigh scattering by entropy fluctuations (WEHNER and KLEIN, 1972). In the static limit, the difference between the isothermal and adiabatic susceptibility is an effect due to the collisions between the thermal phonons. The theory which describes these effects in terms of multi-phonon processes is the phonon transport theory. The recent developments of the microscopic transport theory were achieved by extensive applications of Green function techniques (HORIE and KRUMHANSL, 1964; KWOK and MARTIN, 1966; SHAM, 1967; GOTZE and MICHEL, 1969; ENZ, 1968; NIKLASSON and SJOLANDER, 1968; NIKLASSON, 1970; KLEIN and WEHNER, 1969; MEIER, 1969; BECK and MEIER, 1970), and have led to an increased understanding of the treatment of anharmonic interactions. The reader is also referred to WEHNER (1975).

b) Normal coordinates and lattice Hamiltonian. The dynamical variables of the lattice vibrations are normal coordinates Q which are linearly related to the atomic displacements u by

u(l)= L x(ll A) Q(A). (34.1) A

Here, I is an index running over all degrees of freedom of the lattice in direct space, and the index A specifies the normal vibrations; the coefficients x(ll A) represent the eigenvectors. The corresponding eigenfrequencies are denoted by mAo For perfect lattices, the complex normal coordinates (BORN and HUANG, 1954) are a very important type of coordinates. They are characterized by a wave vector q and a branch indexj for which the joint index A=(q,j), X=( -q,j) is used. The complex normal coordinates satisfy the relation

Q*(A)=O'Q(X), 0'= ± 1 (34.2)

which implies that

x*(lIA)=O'x(lIA). (34.3)

In these equations 0' accounts for two different signs used in the literature. The positive sign is taken by most authors. In the case of real eigenvectors, however, symmetry properties can be more easily expressed by choosing 0' = - 1 (LEIBFRIED, 1955; LEIBFRIED and LUDWIG, 1961: LUDWIG, 1967). The distinction between A and X is necessary for the complex normal coordinates. Because of time-reversal invariance the modes A and 1 are degenerate, mA =mx. If the normal vibrations are enumerated in a different way, for example, in the case of the normal coordinates of the first kind (BORN and HUANG, 1954) or in the case of local modes, then the distinction between A and 1 is of no interest, and one can drop the bar on A. If this is done, the Kronecker symbol bu', which we use later, can be understood as bU'. For the complex normal coordinates, however, its meaning is defined by Born's delta function, bu' = Ll (q + q') bjr . The detailed specification of the normal coordinates is not important for the form of the equations of motion and the treatment of the interactions.

If the forces acting on the atoms depend only on the positions of the nuclei, one can write the potential in the lattice Hamiltonian as a function of displacements from a lattice configuration which is conveniently chosen as given by the

Sect. 34

equilibrium positions

Basic concepts

H= T+If>,

T=~ I pZ(l) 2 I Ml'

If> = If> (u (1), ... , u(l), ... ).

477

(34.4)

Here, p(l) is the momentum belonging to the degree of freedom I, and Ml is the mass of the corresponding atom.

The simplest potential function ensuring stability IS a positive definite quadratic form

1 8z If> If>z =2" b 8u(l) 8u(l') u(l) u(l'), (34.5)

representing harmonic forces. If the first non-constant term of the Taylor expansion of If> is of this form, it defines the harmonic approximation. In this approximation, the Hamiltonian

(34.6)

leads to a set of independent, harmonic oscillators or to free phonons. The harmonic frequencies and eigenvectors follow from the eigenvalue equations

~ 8U(~)z :U(l') x(l' I 2) = w~ Ml x(ll 2). (34.7)

With the eigenvectors x(ll 2) normalized so that

I Ml x*(ll 2) x(ll A')=bu ', (34.8) I

the Hamiltonian in (34.6) can be written as

H(O) =i I (Q*(2) Q(2) + W~ Q*(2) Q(2)). (34.9) A

From the commutation relations in direct space,

[u(l), u(l')] =0, [p(l), p(l')] =0, [u(l), p(l')] =iMll " (34.10)

one obtains for the quantized normal coordinates

(2W )1/2 A(2)= T Q(2), (34.11)

(2 )1/Z A(2)= ~A ~M1X*(112)u(l), (34.12)

A(2)= e~A tZ ~ x*(ll 2) p(l), (34.13)

and the commutation relations

[A(2), A (A')] =0, [A(2), A (A')] = 0, [A(2), A (2')] = 0" 2w A ib;u'. (34.14)

478 Dynamical theory of interacting phonon systems

Defining phonon creation and annihilation operators by

A(Je) =:0 a(Je) + aa + (A),

A(Je) =:0 - iw)Ja(Je) - aa+ (A)), or

1 ( i.) a(Je)=:o- A(Je)+~ A(Je) , 2 w A

a+(Je)=:o~ (A(A)-~ A()~)), 2 w A

it follows that these satisfy the commutation relations given by

[a (Je), a(Jc')] = 0, [a + (Je), a+ (Jc')] = 0, [a(Je), a+ (Jc')] = b U'"

Sect. 34

(34.15)

(34.16)

(34.17)

(34.18)

(34.19)

Using the above relations, the Hamiltonian H(O) can be put into the familiar form of a set of independent harmonic oscillators:

H(O)=~ L (J... A+(Je) A(Je)+wAA+(Je) A(Je)) 4 A w A

(34.20)

= h L w A(a+ (Je) a(Je) +!), (34.21) A

where a + (Je) a(Je) is the phonon number operator. In the harmonic approximation, one can discuss basic properties of crystal

lattices in the picture of free phonons with infinite lifetime. However, in order to explain effects like thermal expansion, the finite thermal conductivity and the measured spectral widths of the phonons, this approximation is insufficient. The expansion of realistic lattice potentials into displacements contains anharmonic interaction terms of higher than the second order. In some cases they can be treated as a perturbation, in other cases they are of fundamental importance, especially when the second-order potential t1>2 is not positive and leads to imaginary phonon frequencies. Furthermore, if static electric fields or static deformations are applied, then a potential term linear in the displacements appears in the Hamiltonian (BORN and HUANG, 1954). Such terms give rise to new equilibrium positions. We consider therefore a general potential function

00

(34.22)

expanded about a convenient initial lattice configuration:

(34.23)

A series expansion like this is only adequate when the potential has no serious singularities. The expansion coefficients are generally dependent on the positions of the initial configuration. Convenient choices of this configuration are defined

either by the classical minimum condition, a:7z) =0, or by the vanishing of the

thermal expectation values of the displacements, i.e. < u (/) > = 0. In the latter case


the expansion coefficients, i.e. the force constants, depend, via the actual volume of the crystal, on the temperature. The second-order term tP2 then defines the quasi-harmonic approximation of LEIBFRIED and LUDWIG (1961). If the equilibrium configuration is not known, one has to start from a trial configuration and calculate the expectation values <u(l). For an ideal lattice, this problem involves a relatively small number of unknown parameters, namely the cell parameters.

In the general case, normal coordinates are introduced as in (34.1) and (34.11), where the displacements are expressed as

( h )1/2 u(l) = f x(ll A) 2m A A(A). (34.24)

The eigenvectors and eigenfrequencies in this equation can be understood as defined in either the harmonic or quasiharmonic approximation, but also as quantities which are determined later according to the requirements of selfconsistency. In the latter case, the precise meaning of the normal coordinates depends on the types of approximations. It should be emphasized that, also in the general anharmonic case, the basic idea is to represent the motion of the atoms in terms of (renormalized) harmonic oscillators. If excitations of approximately this type exist, one can still maintain the familiar picture of phonons.

In terms of the A(A), the n-th order potential (34.23) is written

h tPn=, L tPn(Al> ... ,An)A(A1)···A(An),

n. Al ... An (34.25)

where the coupling coefficients

depend on the eigenvectors and frequencies, which have to be calculated from the lattice potential.

c) Equilibrium correlation and Green functions. Theoretical knowledge about the physical properties of a system can be related most directly to experimental data if it is represented in form of expectation values of dynamical variables. For a crystal lattice, we choose as variables the normal coordinates A(A) and their products. The general relations derived later in terms of these variables can also easily be written in terms of the displacements u in direct space.

At finite temperatures, the thermal expectation value of a quantity A is defined as an ensemble average using the density matrix of the crystal in thermal equilibrium:

<A)=Tr{pA}, where

1 {3= kT'

(34.27)


In the following discussion, three expectation values of this type are considered.

(a) Correlation functions. The thermal expectation value of the product of n Heisenberg operators, which are explicitly written as

A(k) == A(Ak' tk)= e*Htk A (Ak) e -*Ht\ (k = 1, ... , n),

defines the n-time correlation function, denoted by I n ,

I n(1, ... , n) = <A(A1' t 1) ... A(An' tn)·

(34.28)

(34.29)

For n = 1, (34.29) represents the static equilibrium value of a single normal coordinate, i.e.

J1 (At) = <A(A). (34.30)

This quantity determines the equilibrium positions of the atoms and, in particular, is suited to describe structural transitions (PYTTE and FEDER, 1969; FEDER and PYTTE, 1970).

(f3) Time-ordered many-time Green functions. In addition to the correlation functions I n, it is convenient to consider special linear combinations of them. First, it is easier to work with functions which are symmetric in the arguments. The correlation functions of (34.29) are not symmetric, because the normal coordinates commute only for equal times, t 1, ... , tn. Second, there is an essential advantage in using such linear combinations which are Green functions with respect to the differential equations of motion of the correlation functions. Functions which have both properties are the time-ordered many-time Green functions

(34.31)

Here, T denotes a time-ordering operator which can be explicitly represented by the sum over all permutations of the n normal coordinates

Gn(1, ... , n) = L en(tal , ... , taJ I n(a 1, ... , an), Pn

( 1, ... , n ) P,. = . (34.32) a1, ... , an

In this equation, the many-time unit-step function en is equal to one for the permutation with tal> ... > tan and zero otherwise; it can be represented as

(34.33) where

{1,

e(t)= 0,

t>o t<O, e(t)=b(t).

(34.34)

From the time-ordered Green functions all dynamical properties of phonon systems can be deduced. The calculation of these functions will be discused below in detail. In particular, it will be shown how they enable the many-time correlations to be systematically reduced to the picture of interacting phonons.

(y) Double-time Greenfunctions (ZUBAREV, 1960). If in (34.29) and (34.31) one sets t 1 = ... = ti = t, ti+ 1 = ... = tn = t', and abbreviates the products A(Al)··· A(AJ =A and A(Ai+l) ... A(An)=B, one obtains functions which depend only on two


times, actually only on the difference t - t', since the Hamiltonian is assumed to be time-independent. One can therefore put t' =0 and write

Jt(A I B) = <A(t) B(O». (34.35)

Such correlation functions are important in scattering theory (VAN HOVE, 1954), and are related to scattering rates. In the same manner, a time-ordered or causal Green function can be derived from (34.31) with

G~(A I B) = <TA(t) B(O»

= B(t)<A(t) B(O» + B( - t)<B(O) A(t». (34.36)

As a third double-time function the retarded Green function can be introduced with

G;(A I B)=i B(t)<[A(t), B(O)])

=i B(t)«A(t) B(O» - <B(O) A(t»), (34.37)

which is the linear response function (KUBO, 1957). Most experimentally measured quantities are directly related to a correlation

function or a retarded Green function as in (34.35) or (34.37), respectively. Both functions can be calculated from the time-ordered Green function (34.36), which is more basic from the viewpoint of the methods presented here. The various general relations between these double-time functions are considered in detail in Sect. 34e.

d) Double-time Green functions: Harmonic approximation. In case of anharmonic interactions, the correlation and Green functions, defined in the foregoing section, cannot be calculated exactly. There is thus the problem of developing physically appropriate approximations. In many cases the behavior of the general functions is not very different from their characteristic behavior in the absence of anharmonic interactions. The simple properties of the Green functions in the harmonic approximation are often reflected in general expressions and complicated solutions.

In the harmonic approximation, where we have a set of independent harmonic oscillators, the correlation and Green functions can be calculated exactly. With the Hamiltonian of (34.21) and the commutation relations from (34.19), one obtains the equations of motion given by

.1 0 -A(2)= ih [A(2),H( )] = -iw;.(a(2)-aa+(2» (34.38)

and

A(2)= -wi A (2). (34.39)

These equations follow also from the Heisenberg operator A(2, t), which is derived from the definition, (34.28), by commutator algebra, yielding

(34.40)


Elementary quantum statistics applied to harmonic oscillator ensembles (BORN and HUANG, 1954) leads to the thermodynamic expectation values:

and

where

<a(A)(O) = 0, <a+ (A)(O) =0, <A(A)(O) =0,

<a+(A) a(X)(O)=n,\bu ',

f3hw 2n,\+I=cothT ·

(34.41)

(34.42)

(34.43)

(34.44)

With the relations given above and with A = A(A) and B = A (A'), the double-time functions of (34.35) to (34.37) can be written explicitly:

and

J~O)(A(A)I A(X)) = (J J~O)(t) bH'

Jl°)(t) ==Jt(O)(A(A)IA +(A)) = (n,\ + l)e- iw;.t + nA eiw;.t,

G~,(O)(A(A)I A(X)) = (JGi,(O)(t)b H ,

Gi,(O)(t) == G~,(O)(A(A)IA +(A)) =(2n,\ + 1) cosw,\t - i sin wAltl

G~,(O)(A(A) I A(A')) = (JGi(O)(t) bH "

Gi(O)(t) = G;,(O)(A(A) I A +(A)) = 2 e(t) sin w At.

(34.45)

(34.46)

(34.47)

The physical interpretation of the retarded Green function is obvious, as it represents the motion of a harmonic oscillator which is at rest for negative values of the time t and which is excited at time zero within an infinitesimally short period. The correlation and Green functions of (34.45) to (34.47) depend on two normal coordinates of phonons at two different times. Nevertheless, they are called one-phonon correlation and Green functions, respectively, since they describe the creation, propagation, and annihilation of one and the same phonon. More formally, they are called two-time or two-point functions.

For actual calculations one uses the time Fourier transforms of the functions considered so far. The Fourier transform of the correlation function,

00

J~O)(w)= lim S dteiwte-eltIJl°)(t) 13-0+ -00

(34.48)

is the spectral density function. Similarly, the Fourier transforms of the Green functions are

Gi'(O)(w)= lim i{ n,\+I. + n,\. e~O+ w-w,\ +18 W+W,\ +18

and

Gi(O)(w)= lim 2 2w,\. 2' e~O+ W,\ -(W+18)

(34.49)

(34.50)


These expressions represent all dynamical properties of single, free, harmonic phonons in a crystal at finite temperature. The corresponding two- and threephonon Green functions can be calculated in a straightforward way. The expressions for the retarded functions are given later by (38.19) and (38.23).

e) Double-time Green functions: Spectral representations. Some general properties of the correlation and Green functions follow immediately when they are written explicitly in terms of matrix elements Anm and Bnm , where nand mare the exact eigenstates of the lattice Hamiltonian, Hnm = hWn (jnm. In this representation, the density matrix, p, in (34.27) is diagonal: Pnm = Pn(jnm' Pn = e-fJFW)n/Z, and the time-dependence of the Heisenberg operators, (34.28), is simply expressed: A(t)nm=Anmeironmt, wnm=wn-wm. Taking the Fourier transform of the two parts of the time-ordered Green function, (34.36), we obtain

J dt ei(ro+ie)t<A(t) B(O) = i L Pn. A B , o .. nm W+ Wnm+ 18 nm mn

(34.51)

and

(34.52)

As a function of complex frequency w, these two expressions are analytic functions in the upper and lower half-plane, respectively. Generally, the poles are distributed quasi-continuously along branch cuts near the real W axis. Using 8=0 in (34.51) and (34.52), we define two functions with the poles, i.e. the branch cut, on the real axis, but which are each analytic in the upper as well as in the lower half-plane

(34.53)

G;(AIB)= -i L Pm AnmBmn, nmW+Wnm

=G~ro(BIA). (34.54)

The main difference between the Fourier transforms in (34.51) and (34.52) and the functions G± is that the former are analytic functions on the real W axis, whereas the latter are not. The Fourier transform of the time-ordered Green function is recovered from boundary values of G+ and G-, (8-+0+)

G~(AIB)=G:+ie(AIB)+G;_iiAIB)

." (Pn Pm) A B =1l..J . . nm mn° nm W+Wnm +Z8 W+Wnm -Z8

(34.55)

Comparing this expression with the one-phonon Green function in the harmonic approximation, (34.49), we find

Gi'(O)(W) = ±1 ~+--;.-.{l+n n} w+w;. w±w;.

(34.56)

as special cases of (34.53) and (34.54). The most important property of the functions G ± is that the boundary

values from the upper and lower half-planes are different. In case of G+, one


obtains for the difference the so-called spectral-density function

G~+i.(A IB) - G~_i.(A IB) = 2n L PnAnmBmn o(w + wnm) nm

= J",(A IB),

Sect. 34

(34.57)

which is the Fourier transform of the correlation function, (34.35). Similarly, the difference of the boundary values of G- is given by

G;;;-+i.(AIB)-G;_i.(AIB)= -2n LPmAnmBmnb(w+wnm) nm

(34.58)

Because P"'/ Pn = efJ~"'n~, this quantity is related to the spectral-density function of (34.57) by

(34.59)

The Fourier transform of the retarded Green function follows from (34.37) as

G~(AIB)=-L Pn-Pm. AnmBmn nm W + W nm+ 18

=i(G~+i.(AIB)+ G;;;-+i.(AIB)). (34.60)

The expressions in (34.55) and (34.60) for the causal and retarded Green functions are usually written as integrals over the spectral-density function. Using (34.57) and (34.59), the summations over the states nand m can be converted into the integrals

i 00 (1 e - fJ~u ) G~(AIB)=-2 J du . . Ju(AIB),

n -00 W-U+18 W-U-18 (34.61)

1 00 1-e-fJ~u G~(AIB)= -- J du . Ju(AIB),

2n -00 W-U+18 (34.62)

which are the spectral representations. If the time-ordered Green function is explicitly given in a form like (34.55) or

(34.61), G+ + G- can be reconstructed as an analytic function for complex w, and the retarded Green function can be derived in accordance with (34.60). In this sense, we write

(34.63)

This procedure can be better understood by noticing that for the function GC the step from the real W axis to W + i17 is equivalent to increasing GC by

G~+iq(AIB) -G~(AIB)= G~+iq(AIB) + G;;;-+iq(AIB)

-(G~+i.(AIB) + G;_i.(A I B)),

= -J_",(BIA), (34.64)

which follows from (34.55) and (34.58). Instead of (34.63), we can therefore write

G~(AIB) = i {G~(A IB) - J _",(B I A)} , (34.65)


where W is on the real axis. By taking the Fourier transform, the corresponding time-dependent relation

G~(A IB) = i {G~(A IB) - (B(O) A(t)} (34.66)

is obtained in agreement with (34.35) to (34.37). The inverse procedure of deriving the time-ordered Green function from the

retarded one generally does not exist, as has been noted by STEVENS and TOOMBS (1965). If the retarded Green function is given, for example in the form of (34.60), it is not possible to reconstruct G+ and G- because (34.60) contains no information about products of matrix elements AnmBmn for which wn=wm. This means that Gr(AIB) does not depend on the diagonal elements of the operators A and B, nor on matrix elements between degenerate states. The occurrence of such matrix elements is related with a time-independent constant in the correlation function Jt(AIB) or, in the spectral-density function, with a delta function at zero frequency. From (34.57), we obtain

J",(AIB)=2nCo(AIB) c5(w) +J~(AIB), (34.67)

Co(AIB)= LPnAnmBmn, (wn=wm) nm (34.68)

and

(34.69) nm

If (34.67) is inserted into the spectral representations (34.61) and (34.62), the first term contributes to the causal but not to the retarded Green function. The latter depends only on the dynamical part J' of the spectral density function.

The retarded Green function is suited for determining the dynamical part J' of the spectral-density function. From (34.57) and (34.58), we find for the difference of G + + G - across the real axis that

L1G w(A I B) == {G ;;;+iq(A I B) + G;;;-+ iq(A I Bn - {G ;;;_iq(A I B) + G;;;-_iq(A I Bn

=(l-e-P~"')Jw(AIB). (34.70)

On the other hand, with (34.60), we arrive at

L1G",(AIB) = -i{G~(AIB)-G~",(BIAn. (34.71)

If the products of matrix elements AnmBmn are real, as in most cases, then

Gr_",(BIA) = G~(AIB)*, so that

L1G",(A IB) = 2 1m G~(A IB).

Comparing (34.70) and (34.73), we obtain the relation

(1- e-P~"') J",(A IB) = 2 1m G~(A IB),

from which, after inserting J",(AIB) from (34.67), we conclude that

J~(AIB)=2[1 +n(w)] ImG~(AIB), (w*O).

(34.72)

(34.73)

(34.74)

(34.75)


This relation can be used to derive the expressions

1 en

<AB) = Co(AJB)+- S dw[1 +n(w)] 1m G:(A JB), n -en

(34.76)

and G~(AJB)=2n Co(AJB) 6(w)

+~ S du (1+n(W? n -en W-U+lI;

n(w) ) --'------'---. ImG:(AJB). w-u-u;

(34.77)

In many cases, Co is equal to zero, so that J =J', and the retarded Green function is sufficient for calculating <AB) and G~(AJB). If, in addition, the products AnmBmn are not only real but are also symmetric in nand m, as in the case of the one-phonon Green functions where A = A (Il) and B = A + (Il), then the spectral representations reduce to

i en 1 1 G~(w)=-2 S du ( . . )J~(U),

n -en W-U+lI; W+U-18

1 en 2u G~(w)=-2 S du 2 ( . )2 J~(u),

n -en U - W+18 with

nm

It is easily verified from (34.78) and (34.79) that

G~(-W)=G~(W),

G~( - w) = G~(w)*,

(34.78)

(34.79)

(34.80)

(34.81)

(34.82)

and also that the retarded Green function obeys the Kramers-Kronig relations

1 00 1m Gr(u) ReG~(w)=-P S du .Ie,

n -en u-w (34.83)

(34.84)

The latter follow from the Cauchy integral around the upper half-plane where Gr is analytic. Furthermore, relation (34.76), which is known as the fluctuationdissipation theorem, can be put into the special form

1 en f3liw <A(Il) A + (Il) =- S dw coth-ImG~(w).

no 2 (34.85)

The general expressions and relations of this section are extensively used in Sects. 37b and 38a to derive the phonon self-energy and the two- and threephonon Green function in terms of the retarded one-phonon Green function or one-phonon spectral-density function, respectively.

Sect. 35 Functional methods 487

35. Functional methods

a) Non-equilibrium Green functions. The Green functions defined and discussed in the foregoing sections are expectation values which represent properties of crystals in thermal equilibrium. Therefore, they are more specifically called equilibrium Green functions. The change in equilibrium quantities under the perturbation of an external field can, within linear response theory, be expressed in terms of equilibrium Green functions, namely by retarded Green functions. If one expresses time-dependent, non-equilibrium quantities to any order as a function of an applied field, functionals of this field are obtained, from which other interesting quantities can be deduced as functional derivatives. The definition of such general expectation values leads to the so called nonequilibrium Green functions and functional techniques. A particular advantage of the latter technique is that the whole set of time-ordered Green functions (34.31) and the corresponding hierarchy of equations of motion can be treated in a comprehensive and systematic way.

An external time-dependent field, j(2, t), acting on the normal coordinates is introduced with the interaction Hamiltonian

H;=h L A(2)j(2, t), (35.1) A

with

j*(2, t)=uj(X, t). (35.2)

We are here using the field j, which is a c-number and is sometimes called a source field, mainly as a device to generate functionals and to apply functional techniques. There exist physically interesting interactions with a Hamiltonian quadratic either in the phonon normal coordinates, or in the applied field, or in both, which we do not consider in (35.1). We assume, however, that interactions of the crystal with static external fields and macroscopic deformation parameters (BORN and HUANG, 1954) are included in the potentials of (34.25).

If j is time-independent, the interaction expressed in (35.1) leads to a total Hamiltonian H + H' without any new specific feature as compared to H. The extra part H' of the Hamiltonian could be included in cP 1 in which the source field results in a variation of cP 1 (2). Following MATSUBARA (1955), the partition function

Z[jJ =Tr {e-P(H+H'J}

can in this case be written as

where the operator S is given by ~

-SdtH'(t) S=Te 0 •

Here the integrand is defined in the interaction picture as

H'(t)=eHt H' e- Ht,

(35.3)

(35.4)

(35.5)

(35.6)


and T, the ordering operator with respect to the argument t, is expressible as

T[H'(t,,) ... H'(t,,)] =H'(t1) ••• H'(tn)' tl > ... >tn. (35.7)

Interpreting t as an imaginary time, the integral in the exponent of (35.5) can be transformed into a path integral in the plane of complex time T = t + it'. Substituting -t'/h for t in (35.5), the operator S can be written as

-~ Sd,H'(,) S=Te hL ; (35.8)

where the integration path L leads from the origin along the negative imaginary axis to -iPh (dotted line in Fig. 35.1 a). For the calculation of equilibrium quantities by thermodynamic perturbation theory, the partition function of (35.4) is the starting point.

In order to define non-equilibrium quantities in the case of a time-dependent source field, the operator S has been chosen by various authors in different ways which we bring together by taking the integration path L, as shown by the solid line in Fig. 35.1 a. In addition we assume j to be analytic for - ph 2= 1m T 2= 0 and to be double-valued (SCHWINGER, 1961; KORENMAN, 1966; CRAIG, 1968), here denoted by j±. Under these circumstances, the time integration 10 (35.8) is defined by

JdTj ... = J dTr ... +JdTr·· .. (35.9) L Lo+L2+L3 L,

The operator T orders a product of operators from right to left according to the order of the time arguments indicated by the arrows in Fig. 35.1 a. With (35.9) in the S operator, the expression (35.4) becomes a generalized partition function

where Z[j]=Z(S)j,

a)

.. ... to 0 L, -(I)' .. Oi

IL'

I

Lo I L2 I I I I I I • to-i~'h -i~'h

b) L+

-(I)~,==========~ .. ~~4F=============~+~ i L I

iL3 I I •

(35.10)

(35.11 )

Fig. 35.la, b. Integration paths in the complex time plane: (a) general path including the case of MATSUBARA (1955), (b) double path, restricted to real time axis


Here, the superscripts ± have been dropped and the repeatedly occurring variable k is understood to include summation over Ak and time integration over tk along L.

If one puts r = j- at the beginning, the contributions from L o, L 1, and L2 to S cancel each other, and <S)j represents the generating functional used in thermodynamic perturbation theory by many authors, for example KADANOFF and BAYM (1962), HORIE and KRUMHANSL (1964), KWOK and MARTIN (1966), GOTZE (1967), and MEIER (1969). In these cases, the Green functions depend primarily on imaginary time arguments.

The extension of the integration to the real time axis is only non-trivial if r and j- can be varied independently. In order to obtain physical results, one has to set r = j- after the variations. In the limit to -+ - 00, the contribution from L3 is assumed to vanish. In this case the Green functions are only considered for real-time arguments, and <S) j is of the form used by NIKLASSON and SJOLANDER (1968) and BECK and MEIER (1970). Then the integration is done only along L1 and L2 which are denoted in this case by L+ and L_, respectively (Fig. 35.1 b). Ifj- is set equal to zero, so that only L+ is actually used, <S)j reduces to the generating functional treated by HOGBERG (1966, 1967), WEHNER (1966, 1967) and KLEIN and WEHNER (1968, 1969). In this case the physical limit is reached with r =0, i.e. for the equilibrium Green functions.

The double integration path, L=L+ +L_, is adopted in the following. By this kind of path one avoids the problem of analytical continuation which arises, if one uses only L3 or L+. Retarded quantities are obtained in a direct manner from functionals defined on the double path.

In order to discuss the functional derivatives of the expression (35.11) in the limit to -+ 00, the operator S is conveniently decomposed as

(35.12) with

(35.13)

The ordering operator T> is the same as in (34.31), while T<, representing the action of T on path L _, leads to the opposite order. In the case where r = j-, the operators S>(t2' t 1) and S«t1' t2) are respectively the time evolution operator of the interaction picture and its inverse,

(35.14)

The functional derivatives of S are determined from the variation of S induced by a variation of the source field

S[j+bj]-S[j]=L Jdt 5:.b(~ ) bj(A, t). (35.15) A L UJ 11., t

Inserting on the right-hand side a b-function-like variation, bj(A,t)=ebukb(t - tk), the functional derivative is obtained as

bS ~ . 5: '(k) =lim - (S[j + bj] -S[j])= -i TSA(k). UJ .-0 e

(35.16)


Here, (Jk is +1 if tk is on the path L+, and -1 if tk is on the path L_. The different sign arises from the change of direction of integration. Where necessary, we denote the path by subscripts ± of the arguments, e.g. k+ or k_, to underscore the meaning of the time arguments.

We define now the generalized Green functions by

(i)n bn .

Gn j1, ... , n)= Z[j] bj(1) ... bj(n) z[j]. (35.17)

More explicitly, the Green functions are of the form in bn

Gn j1 ±, ... , n±) = <S) j bj± (1) ... bj± (n) <S)j'

1 = <S) j < TSA(1 ±) ... A(n±». (35.18)

There are 2n different derivatives of order n which reduce in the limit r = j- to expectation values with a well-defined physical meaning.

From (35.18) for n = 1, one obtains using (35.12), 1

G1,j(1+)= <S)j <S« - 00, oo)S>(oo, t 1)A(A1, t1)S>(t1' -(0». (35.19)

Putting r = j-, one has (35.20)

where AH(1) is the Heisenberg operator of the normal coordinate A(A 1), defined with respect to the total Hamiltonian H + H; :

AH(1)=S«-00,t1)A(A1,t1)S>(t1' -(0). (35.21)

The thermal averaging is done in (35.20) with the density matrix at t = - 00,

where the crystal is assumed to be unperturbed and in thermal equilibrium. Taking the time t1 on the path L_, one finds

1 G1jC)= <S)j <TS« - 00, t 1) A(A1' t1) S«t1' (0) S>(oo, - (0». (35.22)

In the physical limit, the same expectation value as in (35.20) is obtained so that, independently of the path, we can write

G1,P)=<AH(1», U+ =j-). (35.23)

By using the expansion of this expectation value to first order in the source field, 00

<AH (1» =<A(A1» - L J dtie(t)<[A(A1, t), A(A, O)])j(A, t1 -t), (35.24) A - 00

the retarded Green function is obtained in agreement with the linear response theory (KUBO, 1957).

In the following, we will be mainly concerned with the higher order manytime derivatives which can be written as a 2 x 2 matrix for n = 2

G2 .(1 2)= [G 2 ,j(1+, 2+) G2 j 1+,2_)]. oJ' G2 ,j(C,2+) G2 jC, 2_)

(35.25)


After setting j+ = j- in (35.25), we find that G2 represents the following four double-time functions:

_ iF (S) j _ > H H G2.j(I+,2+)--bj+(1)bj+(2)-(T A (1)A (2), (35.26)

G (1 2 ) __ J 2 (S)j 2.j +, - - Jj+(1) Jj-(2) (AH(2) AH(1), (35.27)

J 2 <S)j G2 , /L, 2+)= - Jr (1) Jj+ (2) <AH(I) AH(2), (35.28)

and

G (1 2 ) __ J 2 (S)j 2.j -, - - Jj-(I) Jj-(2) <T< AH(I)AH(2). (35.29)

Here, on the right-hand side, the path is no longer specified; the time arguments vary from - 00 to + 00 in the usual sense.

In (35.24) the term linear in the source field implies that, because of j+ = r, the retarded Green function is obtained as the functional derivative of G 1 by varying j+ and r both in the same way. This is also reflected by the first row of the matrix in (35.25) where the difference in the two elements leads to

i{G2,/1+, 2+)-G2 • /1+, 2_)} =i8(tl -t2)([AH (I), AH(2)]). (35.30)

This equation is comparable to (34.66) and defines a generalized retarded Green function. The so-called nonequilibrium correlation and Green functions, defined by the (35.26) to (35.30), reduce for vanishing source field to functions of the kind defined in Sect. 34c.

b) Generalized thermodynamic potentials and cluster expansion. In order to develop approximations for the equilibrium Green functions of Sect. 34c, it is convenient to proceed from the functionals defined in the foregoing section and afterwards put the source field equal to zero. In this section the time arguments can be anywhere on the general path L; the subscript j will be dropped as long as there can be no misunderstanding since practically all quantities here are functionals.

In addition to the generalized partition function of (35.10), which is the generating functional of the time-ordered, many-time Green functions, two generalized thermodynamic potentials following MEISSNER (1970a) will be considered. The generalized Gibbs free energy is defined by

1 G[j] == --logZ[j]

f3 1

=F- p ljJ[j], (35.31)

with 1

F= --logZ (35.32) f3 ' and

ljJ[j] =log<S)j' (35.33)


Here, F is the free energy of the crystal without source field. For a static source field and with r = j-, to = 0, i.e. with S given by (35.5), the generalized Gibbs function of (35.31) coincides with the Gibbs free energy as defined in statistical mechanics.

The functional derivatives of the field-dependent part Ij; represent a new set of functionals which we call n-point functions and define as

bn bn

(-p) in bj(I) ... bj(n) G [j] = in bj(I) ... bj(n) Ij; [j]

=In. j(l, ... , n). (35.34)

By definition, the n-point functions In' which in the literature are often called cumulants, are symmetric with respect to their arguments. Differentiating (35.33) and using (35.18), one obtains

11(1)=G1(1),

/2(1, 2)=G2(1, 2)-G1(1) G1(2),

and

(35.35)

(35.36)

13(1,2,3) = G3(1, 2,3) - G2(1, 2) G 1 (3) - G2(2, 3) G 1 (1) - G2(3, 1) G1 (2)

+ 2G 1 (1) G 1 (2) G 1 (3), etc. (35.37)

The inverse relations are derived by writing

<S)j=e1jJUI

and by differentiating. Again, with (35.18) and (35.34), one finds

and

G1(1)=/1(1),

G2(1, 2) = 12(1,2) + /1 (1)/1 (2),

G3(1, 2, 3) =11 (1) /1 (2) /1 (3) + /1 (1) 12(2,3) + /1 (2) 12(3,1)

+11 (3) 12 (1,2) +13 (1,2,3), etc.

(35.38)

(35.39)

(35.40)

(35.41)

These relations represent the type of cluster expansion introduced in statistical mechanics by URSELL (1927) and MAYER (1958). From a formal point of view, this expansion only replaces one set of unknown functions by another one. This is meaningful if the higher-order Green functions G n can be reasonably well approximated in terms of a few n-point functions In of low order. In the harmonic approximation, without interactions, the equilibrium Green functions G~~j= ° factorize completely into the two-point function Il~}= 0(/1(0) = liO) = ... = 0). This reduction is a consequence of the fact that the phonons are, in the harmonic approximation, statistically independent quasi-particles. For vanishing source field, the n-point functions are, by definition, as useful as a Taylor series expansion of the Gibbs free energy in terms of an external field which couples to the single phonon modes.

Sect. 35 Functional methods

The general expression for G n is

Gn(1, ... ,n)= L Sf1(1)··.f1(k) kim

(n~k+ 21+ 3m+ ... ) . f2(k+ 1, k+2) ... f2(k+21-1, k+21)

. f3(k+21 + 1, k+ 21 +2, k+21 + 3) ... f3(k+ 21 + 3m-2, k+21 + 3m-I, k +21 + 3m) ....

493

(35.42)

Here, the sum running over k, I, m, ... accounts for all partitions of n into natural numbers. The symmetrizing operator S means that one must sum over the

n! k ! (2 !)I I! (3 !t m! ... different distributions of the arguments 1, ... , n within the

same partition. The numbers of the terms which are involved in the symmetrized partitions can be taken from tables of combinatorial analysis (ABRAMOWITZ and STEGUN, 1965).

For the detailed analysis, a few general relations are useful. From the definitions of (35.18) and (35.34), it follows immediately that

i b <S) bj(n+1) <S)Gn(1,···,n)=Gn+1(1, ... ,n+1) (35.43)

and

(35.44)

Equation (35.43) can be written in the form

Gn+1(1, ... ,n+l)=Gn (I, ... ,n)f1(n+1)+i bj(:+1) Gn(1, ... ,n), (35.45)

which is suitable for the calculation of the cluster expansion of Gn + 1 if the expansion of Gn is explicitly known.

If one of the arguments in (35.42), say 1, has a special role, one can write the cluster expansion in a mixed form where only this special variable always appears in a function fn and where all other variables are collected, as far as possible, in nonfactorized Green functions. According to the definitions of (35.18) and (35.34), one can write

in - 1 bn- 1

Gn(1, ... , n) = <S) bj(2) ... bj(n) [f1 (1) <S)]. (35.46)

Differentiating and expressing only the derivatives of f1 (1) by higher-order, many-point functions one finds, (Go = 1),

n

Gn(1, ... , n) = L Sfk(1, a2, ... , ak) Gn_k(ak + l' ... , an), {aJ =(2, ... , n). (35.47) k~ 1

The sum S over the various distributions of the variables 2 to n consists of

(n-1) k -1 different terms for given k.


The second thermodynamic potential, which will be considered here, is a generalized Helmholtz free energy. This quantity is obtained from the Gibbs free energy through a Legendre transformation (CALLEN, 1960; MEISSNER, 1970a), by which one changes from the sourc;e fiel9 j to the one-point function 11 as an independent variable, . . bG[j]

F[f1]=G[j]-J(k) bj(k) , (35.48)

=F-~ (t/I+ ij(k)11 (k)). (35.49)

In the following, the functional derivatives of the Helmholtz free energy with respect to 11 will be of importance. These derivatives will turn out to represent effective coupling coefficients characterizing interactions between the renormalized phonons. The first derivative follows as

if3 b/(k) F[f1] =j(k); (35.50)

this relation will be used later to eliminate the source field.

36. Phonon dynamics

a) Basic equations 01 motion. The time dependence of any quantity or property of a crystal is determined by equations of motion. To derive such equations for the Green functions Gn and n-point functions In' it is necessary to consider the detailed form of the lattice Hamiltonian, which is given by (34.4), (34.22) and (34.25) as

1 H=T+hL ----. L !PnP·1,···,An)A(A1)···AP·n)· (36.1)

n=1 n. "' ... "n

Here the potential has been expanded about an arbitrary lattice configuration in terms of the normal coordinates of oscillators, which can be considered either as given in the (configuration-dependent) harmonic approximation (!p 1 =!P 3 = ... =0) or as trial oscillators, whose frequencies and eigenvectors have still to be determined by a self-consistent procedure. To develop both points of view, it is convenient to subdivide the Hamiltonian of (36.1) in the following way:

Here, H = H(O) + V. (36.2)

H(O)=T+~ ~W"A+(A)A(A),

=h L w" {a+(A) a(A)+i}

"

(36.3)

(36.4)

is the Hamiltonian of noninteracting oscillators, which formally is the same as (34.21), but can now have a different interpretation as trial oscillators. To account for the auxiliary potential term introduced in (36.3), one has to take

h -V=!p-a"4 ~ W"A(A) A (A). (36.5)

Sect. 36 Phonon dynamics 495

The new potential V, however, can still be written in the form of the (34.22) and (34.25),

(36.6) n= 1

h v,. =, L Vn(A-1'"'' An) A(A1) ... A(An), n. AI".An

(36.7)

where the interaction coefficients are given by

and

(36.8c)

From (34.7) and (34.8), one finds in the initially discussed harmonic approxima

tion that 4>2(A1, A2)=I W AI c51IA2 , i.e. V2(A 1, A2)=0. Under this condition, one

can therefore look upon the Hamiltonian in (36.2) as a perturbation-theory problem or, without this condition, as a problem to be treated self-consistently. Equation (36.8a) provides the link between these two views.

The Hamiltonian of (36.2), the commutation relations of (34.14), and (34.3) together lead to the fundamental equation of motion of the normal coordinates

A(A1, t 1) = -wi. A(Al' t 1)- 2WAI {VI (Ai)

+ L L Vm(Ai,A2, .. ·,Am)A(A2,t1) .. ·A(Am,tl)' (36.9) 00 1 } m=2 (m-1)! A2".Am

Here, A* indicates that the complex conjugate eigenvector has to be taken in the expressions (36.8) and (34.26) for the interaction coefficients. By this notation, one can write most of the equations in a form independent of the choice of (1. In the following, we use the abbreviation

(36.10)

For m~2, the instantaneous interaction coefficients can be written as many-time interactions by multiplying the interaction coefficients with Dirac delta functions:

Vm(1, .. ·, m)= Vm(A 1 ... Am) c5(t1' t2) c5(tl' t3)'" c5(t1' tm),

c5(ti' tk ) = (1ik c5(ti - tk), «(1ik) = (~ _~). (36.11)

The values of (1ik depend on the positions of ti and tk on the double path L; they can be written as a matrix, like that in (35.25). With these conventions, (36.9) can


be put into the form

Ll A(1) = - VI (AT) - m~ 2 (m ~ 1)! Vm (1 *,2, ... , m) A(2) ... A(m), (36.12)

where the pairs of variables 2 to m obey the convention introduced in (35.11). The time integrations are performed along the double path L = L + + L _. Because of the delta functions in (36.11), the time arguments of the normal coordinates A(A2 ) to A(Am) are, after the integrations, all equal to the value of t 1; in particular, they are all on the same path as t 1> either on L + or on L _.

In order to derive the equation of motion of G 1. /1), one differentiates (35.19) and (35.22) twice. With

a :;-SZ(t, +oo)=-iIA(A,t)j±(A,t)SZ(t, +00), (36.13) ut A

a . at sz(± 00, t)=lSZ(± 00, t) I A(A, t)j±(A, t), A

one can show that

a 1. at; G1./1)= <S) <TSA(1),

and

where tl is anywhere on L+ or L_. Employing (36.12), one arrives at

Ll G 1./1)+j*(1)+ V1 (kj')

(36.14)

(36.15)

(36.16)

+ m~2 (m~1)! Vm (1*,2', ... ,m')Gm _ 1,/2', ... ,m')=O. (36.17)

From this functional equation, the equations of motion of the Green functions G n. j are deduced for arbitrary n by operating from the left with

i n - 1 6n - 1

~ 6 6 <S) .... <S) j(2)... j(n)

(36.18)

The resulting equation

. n 6j*(1) L 1 Gn,/1, ... ,n)+1 I 6.(k) Gn _ 2 ,/2, ... ,k-1,k+2, ... ,n)

k~2 }

+ 0*(1) + VI (Ai)} Gn _ 1 • j(2, ... , n) 00 1

+ I ( _ )' Vm (1*,2', ... ,m')Gn+ m _ 2 j2', ... ,m',2, ... ,n)=O (36.19) m~2 m 1.

includes, as a special case, the equations of motion of the equilibrium Green functions of (34.31), if j is put equal to zero and if all time arguments are restricted to the path L+. In particular, one has

(36.20a)


i.e.

(36.20b)

It is characteristic of (36.19) that on the right-hand side only Green functions of order higher than n appear in connection with the anharmonic interactions; these terms and the effects arising from them are here the main point of interest. The cluster expansion, introduced in the foregoing section, enables to some extent a decoupling of (36.19) if only n-point functions of low order are retained. Therefore, the equations of motion of the functions in will be concentrated on in the following.

b) Equilibrium positions. The thermal expectation values of the displace-ments

<u(l) = LX(I[ A) (_I1_)IIZ <A(A) A 2w,,-

(36.21)

define the equilibrium positions. In principle, one has to calculate these static quantities for all degrees of freedom. In case of a free crystal the expectation values <u(l) can only be determined uniquely by excluding rigid translations and rotations. The macroscopic state of deformation, in particular the thermal expansion, follows from the long-range behavior of the expectation values <u(A). The contributing normal modes generally depend on the size and shape of the crystal and their contribution to the long-range behavior cannot be treated in the infinite-crystal approximation with periodic boundary conditions. Homogeneous deformations involve six degrees of freedom of this kind. The corresponding displacement field is usually described in an ad hoc manner by the deformation tensor components eaf3 , which are treated as static variables.

For vanishing source field, it is found from the equation of motion (36.17) that

(36.22)

2 where -=G~(O)(w=O).

W Al 1

Using the relations (36.8) this is equivalent to

1 <P 1 (,11) + m~z (m -I)! "-2 ~Am <Pm(Al' Az, ... , Am)<A(AZ) ... A(Am) = O. (36.23)

This equation is simply expressed in terms of the first derivative of the potential <P; in direct space it is written

(a~(l) <P(u(l), ... , u(i), ... )) =0. (36.24)

This is the general condition for the equilibrium positions.


A simple, formally tractable approximation consists of using for the thermal expectation values the cluster expansion of (35.42) but retaining only the onepoint function (In = 0, n ~ 2). In this case instead of (36.23) one has

or in direct space, as in (36.24), one finds

o ou(l) <P(u(l), ... , u(i), "')u(i)=<U(i» =0. (36.26)

In this approximation all correlations, in particular the thermal and zeropoint fluctuations, are neglected. If the lattice potential is expanded about the equilibrium sites, we have, by definition, <u(i) =0 and (36.26) is simply written

o<P O. ou(l)u(i) = 0

(36.27)

In the basic references for lattice dynamics, BORN and HUANG (1954) and LEIBFRIED (1955), this condition is used to define the initial lattice configuration for developing the perturbation-theory approach. From (36.27) the equilibrium positions follow in the classical limit (h-+O) at zero temperature.

In most cases, one prefers to expand the lattice potential around the temperature-dependent equilibrium positions so that the one-point function can be dropped, i.e. 11 (A) = <A(A) =0. However, to calculate effects of static perturbations which are already included in the potential <P, it is more appropriate to choose as initial lattice positions the unperturbed lattice configuration which may be of higher symmetry.

The approximation made in (36.25) can be improved by admitting two- and three-point functions in the cluster expansion. For low-order anharmonicity one obtains from (36.22) that

<A(A1) =11 (A 1)

= -~ {V1(Ai) + L V2(Ai, A2)/1 (A 2) W A A2

+1 L V3 (Ai, A2, A3 ) [/1 (A 2) 11 (A3) +/2(A2' A3)J

(36.28)

Here, the equilibrium n-point functions

(36.29)

are time-independent. The various terms in this equation are illustrated by diagrams in Fig. 36.1. A rigorous treatment of the one-point functions would require knowledge of the higher-order functions In (n>2). The equations for


~ = , +T+v+Q +

a b c d

+..t..+..}) + tTJ e f 9

• , \) Fig. 36.1. Diagrams of contributions to the one-point function in the case of low-order anharmo

nicity. The vertices, i.e. the interaction coefficients v", are drawn as solid dots

these, however, also contain the quantities f1 as can bc seen in the following sections. This situation demands a self-consistent solution which in general is very difficult.

The perturbation theory approach to (36.28) starts in the harmonic approximation, V2 P'1' ).2)=0, with

f1(0)().) =0, f1°)()., ).')=a(2n;. + 1) Du ', fjO)().l' ).2' ).3)=0, (36.30)

as can be seen from (34.41) and (34.46). To first order in the interactions, the result is

(A().)(1) = fP)().)

= -~ {V1().*)+tI V3 ()'*, X, X*)[2n;.,+1]}, w;. ;"

(36.31)

if higher anharmonicities than V4 are neglected. In view of the treatment of the anharmonic effects in the following sections,

it is convenient to write (36.25) as the condition of vanishing of a renormalized linear interaction coefficient

(36.32)

This new coefficient is obviously the first-order coefficient of the expansion about the positions defined within the approximation of (36.25). The superscript [1] represents the kind of renormalization where, of all the dynamical properties entering through the higher-order anharmonic interaction coefficients, only the trivial one-point function is left. Analogously we write the exact Eq. (36.23) as 4>~OOl().l) =0, where the superscript means that on the left-hand side of (36.23) npoint functions of any order are considered.


c) The renormalized harmonic approximation. A discussion of the phonon excitation spectrum requires the study of the two-point function 12 (1,2) and of the corresponding retarded Green function. In the harmonic approximation, these functions are given by the equations of Sect. 34d. One systematic approximation for calculating the two-point function 12 in case of anharmonic interactions consists in neglecting all n-point functions with n larger than two. Renormalized phonons are defined by the poles of the resulting Green functions, with temperature-dependent frequencies and eigenvectors and with infinite lifetime. This approximation is often called renormalized harmonic approximation (RHA).

The equation of motion of the two-point function follows from the equation

of motion (36.17) of the one-point function by operating with i t5 ~(2) from the left: }

(36.33)

Here we have introduced a renormalized linear interaction coefficient, analogous to expression (36.32):

(36.34)

In differentiating this quantity we assume that the chain rule can be applied, which leads to t5 t5 V[ro](1) t51 (21) . V[ro](1) _ 1 . 1

1 (5j(2) 1 - t5/1 (21) 1 t5j(2) ,

== I' (1, 21)/2 (21,2). (36.35)

The quantity I', defined by

(36.36)

is the so-called self-energy. Using this definition we put the equation of motion (36.33) into the form of a Dyson equation,

Ld2(1, 2) + i t5tj*(~) + I' (1 *,2 1)12 (21,2) =0, (36.37)

from which the two-point function can be calculated. To derive explicit expressions for the self-energy, we put the last term of

(36.33) into the form of the right-hand side of (36.35). Using the recursion formula (35.45) and the mixed cluster expansion (36.47), we obtain

i () [ro]

t5j(2) V1 (1)

ro 1 = I ) , Vm(1, 21, ... , ml){ -11 (2) Gm_ 1 (21, ... , ml) + Gm(2, 21, ... , ml)} m~2(m-1.

ro 1 m

= I ( -1)' Vm (1,2 1, ••• ,ml) L Slk(2,1X2,···,lXk)Gm_k(lXk,···,lXm),

m~2m. k~2 where {IX;} =(21, ... ,ml). (36.38)


Because of the symmetry of the coupling coefficients, we can write this as

i Oj~2) Vl°O](l) = k~2 (k ~ 1)! ~[oo](l, 2', ... , k')lk(2', ... , k', 2), (36.39)

where we have renormalized coupling coefficients:

Vk[OO](1,2, ... ,k)= f ( ~k)' Vm (1,2, ... ,k,k+1, ... ,m)Gm _ k(k+1, ... ,m). (36.40) m=k m .

These new interaction coefficients are still instantaneous, and the Green functions of (36.40) reduce in the equilibrium case to thermal expectation values like those on the right-hand side of (36.22). The equation of motion (36.33) can now be put into the form:

LJ1 (1, 2) +i °1;S/ + k~2 (k ~ 1)! ~[ool(l *,2', ... , k')lk(2', ... , k', 2) =0. (36.41)

Retaining only one- and two-point functions we arrive at the approximate Dyson equation

Here, the self-energy is given in the approximation

1'[2](1,2)= VY](1,2)

_ ~ 1 [2] - f..., ( -2)1 Vm (1,2,3, ... ,m)Gm _ 2(3, ... ,m).

m=2 m .

(36.42)

(36.43)

The superscript [2] indicates that the cluster expansion of G~~ 2 contains only the functions 11 and 12' When used in (36.42) the coupling coefficients VY] represent instantaneous, second-order, effective interactions, i.e. renormalized harmonic forces. This is the reason why this general mean field approximation is called the renormalized harmonic approximation.

For values of m up to 4 we have

1'[2](1,2)= V2(1,2)+ V3 (1,2,3)/1(3)

+1 V4 (1, 2, 3,4)[11 (3)/1 (4)+ 12(3, 4)]; (36.44)

these terms are shown as diagrams a to d in Fig. 36.2. If 11(k)~0, only evenorder coupling coefficients appear in (36.43) with

(36.45)

Here, the sum consists of ~~~i terms. Equation (36.45) corresponds to Wick's

theorem which has been derived for general fields by ENZ (1965). From the solutions of (36.42) one obtains the phonon excitation spectrum

within the renormalized harmonic approximation. A simplifying property of the self-energy of (36.43) is that it is only non-zero if t1 and t2 are on the same path (cf. (36.11». We write the self-energy as a 2 x 2 matrix as was done for G2 in


= + ! a b

Fig. 36.2. Complete set of self-energy diagrams including interaction vertices V. and n-point functions in with n:~ 4

(35.25)

(36.46) 1:[2](1 2)=II[2] b(t -t ) [1 OJ . , Al A2 1 2 0-1

Putting j = 0, the self-energy elements II }.1}.2 are time-independent and given by

II~~]A 2 = VP]P'l Az)

_ ~ 1 '\' [Z] - L- L- Vm(A1 , AZ,A3, .. ·,Am) <A(A3) .. ·A(Am) . m ~ 2 (m-2)! }.3 .. ,Am

(36.47)

Furthermore, the relations (36.20) are equivalent to

bj*(l) [1 OJ bj(2) =rrbIQA2 b(t1-tZ) 0 -1 . (36.48)

With (36.46) and (36.48) we obtain from (36.42) four decoupled equations of motion of Green and correlation functions . In the equilibrium case these four functions are related to those defined in Sect. 34c by

<A(A2 ,t2)A(A1,t1) J < T A(A1, t 1) A(A2 , t 2 )

(36.49)

According to (35.30) the equation of motion of the retarded Green function

GLIc,(t1 - t2 ) = i {fz(l + , 2+) - fz(1 + ' 2_)} U =0)

= G;I_ 12 (A(A1) I A (AZ))' (36.50)

Sect. 36 Phonon dynamics

is obtained by subtracting the corresponding components of (36.42),

L j G~,~)tj -t2)+ 2)I~1\2 G~2A2(tj -t2)=ab;;:, A2 b(tj -t2)· A2

503

(36.51)

For negative time argument the retarded Green function vanishes. Therefore, the Fourier transform of (36.51) leads to

L {[ -(co+i8)2 +coU bA1A2 +2CO A1 JI~;]Az} G~2A2(co)=a2coA1 b;;:,A2' ~ 1

(36.52)

Dropping the self-energy, we find that the zero-order Green function,

Gr,(O)( )_ l' 2CO A1 s: A1A2 co - 1m a 2 ( . )2 U ;;:A2'

"~o+ COAl - CO+18 (36.53)

is the same as the Fourier transform of the time-dependent harmonic Green function, given by (34.47) and (34.50). If (A(A) =0 the corresponding zero-order Green function fiO)(1+,2+) is equal to the expression in (34.46) with t=t j -t2. Using (36.53) we obtain from (36.52) the Dyson equation in the standard form:

(36.54)

The retarded Green functions following from this equation exhibit sharp resonances, which represent the phonon excitation spectrum in the renormalized harmonic approximation.

The derivation of the Dyson equation (36.54) is based on the assumption of a convenient initial set of phonon frequencies and eigenvectors (harmonic approximation or trial oscillators). The actual frequencies and eigenvectors are obtained by performing the unitary transformation, which diagonalizes the matrix of the system of Eqs. (36.54). If one could use the final results of this diagonalization from the beginning in (34.24), one would end up with

(36.55)

The vanishing of the self-energy represents the condition for self-consistency; expressing JIl2] by (36.43) and using (36.8), this condition can be written either as

(36.56)

or in the form

COAl bA1A2=2{:P2(Ai,A2)

+ m~3(m~2)! A3~Am ([>m(Ai,A2,A3, ... ,Am)(A(A3) ... A(Am)l2} (36.57)

The latter self-consistent equation relates the phonon-excitation spectrum directly to the lattice potential. It has to be complemented by several equations, namely (34.26) for the coupling coefficients, and (36.22) for the one-point


function in the appropriate approximation:

<A(A1) = f1 (A 1)

= -~{V1(Ai) w)q

(36.58) + m~2 (m~1)! Vm(Ai,A2, ... ,Am)<A(A2) ... A (Am)[21}

Furthermore, to calculate the expectation values on the right-hand sides of (36.57) and (36.58) we need, besides f1' an equation for the two-point function at equal time arguments:

(36.59)

The first term can, to some extent, be calculated from the retarded Green function by (34.76), which gives rise to

1 00

f2(A 1,A2)=- S dw[1+n(w)]ImG~ A (w)+C 2(A(A 1)IA(A2))· (36.60) n -00 1 2

Here, the term C2 includes the second term of (36.59). It is given by

if matrix elements between degenerate states can be excluded, so that only diagonal elements contribute to Co in (34.76). In simple crystal structures of high symmetry, the diagonal elements and also f1 vanish. In perfect lattices, which are translation ally invariant, diagonal elements A(A),m can occur only for modes with q = 0, which describe macroscopic internal strain. At zero temperature C2 is equal to zero if the ground state is nondegenerate.

In cases where the quantity C2 is nonvanishing, the system of Eqs. (36.57) to (36.60) can be closed only by neglecting this term. C2 can be calculated neither from the retarded Green function G~, ).2' which does not depend on diagonal elements, nor from the equation of motion of the two-point function itself, where the value off2(AuA2) is needed to satisfy the boundary conditions. In general Cz is related to the difference between the quasi-static limit of the retarded response function (35.24) U(t) cx:eBt ), which is sometimes called the isolated response function (WILCOX, 1968; KWOK and SCHULTZ, 1969), and the corresponding isothermal response function. The latter is calculated from the free energy of (35.31) by putting r = j-, so that only the path L3 contributes and by differentiating twice with respect to j. To avoid such differences, one may introduce additional fields into the Hamiltonian (FERNANDEZ and GERSCH, 1967) which enable suitable representations to be chosen, in which no diagonal elements appear. A dynamical treatment of this type will relate C2 to low-lying transitions involving dissipative effects. Within a harmonic approximation, where any damping of the phonons is neglected, the quantity C2 is consequently dropped.

d) Dyson equation with dispersive interactions. Within the renormalized harmonic approximation discussed up to now the properties of anharmonic


crystals can be treated in terms of phonon excitations with infinite lifetime. The one-phonon spectral density functions have so far been sharp delta functions. This situation changes when the basic restriction of retaining only the one- and two-point functions is dropped. Including the three- and four-point functions in the cluster expansion we obtain additional self-energy terms, which can be interpreted in terms of phonon decay processes and collisions between phonons. These interaction processes, leading to finite phonon lifetimes, generally describe irreversible phenomena. They are often called dispersive interactions because their contributions to the self-energy are no longer characterized by a delta function time dependence as in (36.46) but extend over finite time intervals. As a consequence, the Fourier transform of the self-energy will no longer be a constant but a function of frequency.

Improving the renormalized harmonic approximation we may, as a first step, include the three-point function in the equation of motion (36.42). From (36.41) we obtain

Ld2(1,2)+i (y;(~\) + VP1(1 *,2')f2(2',2)

+1 VJ31(1 *,2', 3')f3(2', 3',2) =0. (36.62)

Here, the superscript [3J means that the renormalized coupling coefficients

Vk[3 1(1, ... , k) = ~dm ~ k)! Vm (1,···, k, k + 1, ... , m) G~~k (k + 1, ... , m) (36.63)

depend only on one-, two-, and three-point functions but are not restricted in the order of anharmonicity. Equation (36.62) can be written like the Dyson equation (36.37) by separating a two-point function from f3. We assume that the three-point function,

f (1 2 3)=i c5f2(1,2) 3 ' , c5j(3)

= c5 f2 (1, 2)f (3' 3) c5fl(3') 2 , ,

(36.64)

can be calculated by applying the chain rule as in (36.35). In the new function c5 f2 (1, 2)/c5fl (3') the arguments 1 and 2 have the meaning of phonon coordinates whereas 3' is effectively a variable like that of an interaction coefficient. The time argument t'3 varies along the full path L. Inserting (36.64) into (36.62) we obtain the self-energy

c5f (2' 3') 2:[31(1 2)= V[31(1 2)+lV[31(1 2' 3') 2 ,

, 2' 2 3 " c5 fl (2) . (36.65)

Including the four-point function we differentiate (36.64) further

f4(1, 2, 3, 4) = i c5j~4/3 (1,2,3)

, c5 { , c5f 2 (1,2)} =f2(4,4)c5fl(4') f2(3,3) c5fl(3') . (36.66)


This expression leads to the self-energy

bf (2' 3') 1'[4](1 2) = V[4](1 2) +.1 V[41(1 2' 3') z ,

, z, Z 3 " 15 fl (2)

+.1V[4](1 2' 3' 4,)_b_{bf z(2 1,3 1)f (4" 4')} 64 ", bfl(2) bfl(4") z , .

(36.67)

By straightforward differentiation one can generate the higher many-point functions and perform a series of steps towards the limit of the complete selfenergy 1'[001(1,2). If only cubic and quartic anharmonicities are considered, the renormalized interaction coefficients are trivially given by

vjn1(1, 2) = Vz(l, 2) + V3(1, 2, 3)fl (3)

+1 V4(1, 2, 3,4) [fl (3)fl (4) + fz (3,4)],

V1n1 (1,2,3)=V3(1,2,3)+ V4(1,2,3,4)fl(4), n~l

n~2 (36.68)

VJn1(1, 2,3,4) = V4 (1, 2, 3, 4), n ~ 1.

The instantaneous self-energy of (36.46) turns out to be zero if t 1 and t z are on different paths. This restriction no longer holds if dispersive interactions are taken into account. In order to discuss this more general case we write the Dyson equation (36.37) in matrix form using a notation as in (35.25)

(36.69)

In the last term the product of two matrix elements implies time integration along either L+ or L_.

From the system of coupled equations (36.69) one immediately derives the equation of motion of the retarded Green function of (35.30). Putting r = r one obtains first with (35.40)

Gj(1,2)=iEJ(tl -tz)<[AH (1),A H (2)])

= i {fz (1 + ' 2 +) - fz (1 + ' 2 _)}. (36.70)

In this equation, the one-point function is cancelled out because fl (1 +) = fl (L). If one varies simultaneously j+ and r in the latter relation, one finds furthermore

fz(1+, 2+) - fz(l +' 2J-fz(L, 2+) +fz(L, 2_)=0, j+ =j-. (36.71)

Using (36.70) and (36.71) one arrives, by subtracting the two elements of the first row in (36.69), at

00 LIGj(1,2)-abxlA2b(tl-tZ)+ S dt~{1'(1!,2'+)-1'(1!,2'_)}Gj(2',2). (36.72)

-00

By the expression in brackets the retarded self-energy is defined

1'(1+,2+)-1'(1+, 2_)=n(A j , t1 ; Az, t z). (36.73)


In the equilibrium case, j=O, the retarded Green function and the self-energy depend only on time differences, and we can write (36.72)

L1 G~IA2(t1 -tZ)=()'()AI A2 <5(t1 -tz) 00

-I S dt~IIxiA,Jt1-t~)G~2A2(t~-tz)' ..1.2 ~ co

(36.74)

The Fourier transform of this equation is the Dyson equation in the same form as in (36.54) but with the general frequency-dependent self-energy:

G~IA2(W)=Gil(~;(W)- I Gil(~}(w)IIA\A2(w)G~2A2(w). (36.75) A'1 A2 1

Here, the zero-order Green function is given by (36.53). The calculation and the typical effects of the frequency-dependent self-energy are discussed later on.

The Dyson equation is the key equation for the calculation of the phonon excitation spectrum. If we intend to describe the dynamical properties of a lattice in terms of a phonon quasiparticle picture, we have finally to reduce all higher-order Green functions and n-point functions to the two-point function. This has been done to some extent by cluster expansion and by expressing in (36.64) and (36.66) the three- and four-point functions by derivatives of 12 with respect to the one-point function. But this form of the self-energy is still not suitable for calculations. Only by an explicit analysis of the derivatives appearing in (36.67) in terms of the two-point function can we obtain a system of equations which may be resolvable.

To discuss the self-energy in more detail, we consider the equation of motion of the three-point function13 which can be derived in two slightly different ways. Firstly, writing <5

Ld3(1,2,3)=L1i <5j(1/z(2,3)

, <512(2,3) =Ldz(l,l) <511(1')' (36.76)

we obtain with the Dyson equation (36.37) and with (36.20)

L 1(1 2 3)={_i<5j*(1) -1:(1* 2')1 (2' I')} <512(2,3) 1 3 , , <5 j (1') , z, <511 (1 ')

. <512(2,3) =-l(J <511(1) 1:(1*,2')13(2',2,3). (36.77)

On the other hand, we arrive from <5

Ld3(1, 2, 3) =L1 i <5j(3/2 (1, 2)

= 12(3,3') <5/(3') Ldz(l, 2)

with (36.37) and (36.36) at <5

Ld3(1, 2, 3) = -12(3,3') <511 (3') {1: (1 *,2')12(2', 2)}

(36.78)

_ <5 2 Vlool(l *) , , *" - - <511(2')<511(3'/2(2,2)12(3,3)-1:(1 ,2)13(2,2,3). (36.79)


Comparing Eqs. (36.77) and (36.79) and defining

02 V[ool(l) 011 (2) Oil (3) 1; (1,2,3),

we find the relations

and

012(2,3)

0/1(1) - i 1; (1,2',3')/2(2',2)/2(3',3),

13(1,2,3) = - i/2 (1, 1')/2(2,2')12 (3,3') 1;(1',2',3').

Sect. 36

(36.80)

(36.81)

(36.82)

The quantity r3 is the so-called cubic vertex part. It represents a generalized noninstantaneous cubic interaction coefficient. The vertex part is a symmetric function of its arguments. This follows from differentiating (36.17), which we write using (35.50)

o -VlOO1(1) = -ifj Oil (k) FU1] - CT Ltfl (1).

From this relation we obtain

and

By continued differentiation, higher-order vertex parts can be defined

on r;.(l, ... ,n)= -ifj 0/1(1) ... o/l(n) FU1]' n>2,

on-2 ~-:-::-:------::-::-:-:-E (1, 2), 011 (3) ... 011 (n)

o = O/1(n)rn - 1(1, ... ,n-1).

(36.83)

(36.84)

(36.85)

(36.86)

(36.87)

(36.88)

From the self-energy expression (36.44) it can be seen that in lowest-order approximation the vertex parts are equal to the anharmonic interaction coefficients:

J;<°)(1, 2, 3) = V3 (1, 2,3)

r,(O) (1. 2. 3.4) = V. (1. 2. 3.4).

(36.89)

(16901

Sect. 37 Vertex renormalization 509

Introducing the vertex parts into the self-energy, we restrict ourselves to the approximate expression (36.67), (fn = 0, n ~ 5), for which we find

E[41(1, 2)= Vi41(1, 2)-~ V1 41 (1 , 2', 3')/2(2',2")/2(3',3")1;(2",3",2)

-~ V[41(1 2' 3' 4')f (2' 2")f (3' 3")f (4' 4")[', (2" 3" 4" 2) 64 '" 2' 2' 2' 4 '"

-t VJ41(1, 2', 3',4')/2(2', 2")/2(3', 3")

. [; (2" 3" 5')f (4' 4")f (5' 5") [; (4" 5" 2) 3 " 2' 2' 3"· (36.91)

The second, third and fourth terms on the right-hand side represent the dispersive interaction process. They are shown as diagrams e to h in Fig. 36.2, assuming only cubic and quartic anharmonicity, so that the Eqs. (36.68) hold.

The vertex parts r.. can, according to the relations (36.87) and (36.88), be represented in terms of the two-point function, renormalized interaction coefficients v,,[ml and higher-order vertex parts. This leads to a system of coupled integral equations which, however, can be treated explicitly only within approximations.

37. Vertex renormalization

a) Vertex part ihtegral equations. The renormalization of the cubic and quartic interaction coefficients has been introduced here in two qualitatively different steps. In Sect. 36 c the renormalized instantaneous interaction coefficients V!ool have been defined. The more general dynamical interaction coefficients r.. (vertex part functions) have been obtained in Sect. 36d by including dispersive interaction processes. To cover most of the theories developed so far in the literature, it is sufficient to treat the vertex parts in the following three quantities. Firstly, one has to consider the corrected bubble diagrams e and f of Fig. 36.2, i.e. the second term of the self-energy expression (36.91)

E(l, 2)corr. bubble = -~ V141 (1,2', 3') 12 (2',2") 12 (3', 3") r3 (2", 3", 2). (37.1)

By this expression, which is shown diagrammatically in Fig. 37.1 a, the cubic vertex part r3 enters the Dyson equation and the two-point function in the simplest way. The time-ordered three- and four-time Green functions depend explicitly on the vertex parts in the following way (/1 =0):

G3 (1,2,3) = - i/2 (1, 1') 12 (2,2') r3 (1',2: 3') 12 (3', 3), (37.2)

and

G4(1, 2, 3,4) =/2(1,2)/2(3, 4)+ G~F(l, 2; 3,4)

-12 (1, 1') 12(2,2') 13 (1',2', 1") 12 (1': 1''') r3 (1": 3',4') 12(3', 3) 12(4',4)

_GHF(l 2'1' 2')[; (1' 1"3')f (1" 1"')[; (2' 1"'4')f (3' 3)f (4'4) 4", 3" 2' 3,,2' 2'

- i/2 (1, 1') 12 (2,2') r4(l', 2', 3',4') 12(3', 3) 12 (4',4). (37.3)


a <=l>

b ~=J>-

c 'w' = I l+{=+>;}+ + ~ +

+ {=+X}r +

+==-:= -f2

[-J ~V3 ~r3 tlJ 4

Fig. 37.la-c. Bubble diagram (a) and the three- and four-time Green functions (b and c) involving the vertex part functions

Here, (37.4)

denotes the four-time Green function in the Hartree-Fock approximation, which describes the propagation and exchange of noninteracting pairs of phonons. The Eqs. (37.2) and (37.3) are given in diagrammatic form in Figs. 37.1 band 37.1 c.

The vertex parts are derived, according to (36.87), by differentiating the selfenergy with respect to the one-point function. The results are integral equations, which are considered here within approximations suited for microscopic foundations of the phonon transport theory and of Landau's theory of phonon quasiparticles and their interactions. For an account of the phenomenological theory, the reader is referred to GOTZE and MICHEL (1967b).

The differentiation of the two-point functions and of the vertex parts r3 and r4 in (36.91) is straightforward. The derivatives of the renormalized instantaneous interaction coefficients

~f(j I vt"'I(1, ... , k) u 1 ()

= I 1 (j __ " (m-k)! Vm(1, ... , k, k+ 1, .. . , m) f>f, (l) Gm_k(k+ 1, ... , m) (37.5)


depend on the derivatives of the time-ordered many-time Green functions Gn •

We calculate the latter by using the chain rule

. b ( ) bGn(l, ... , n) f (1' 1)' 1 b j (1) G n 1, ... ,n = b il (l') 2" (37.6)

For this expression we obtain, with the recursion formula (35.45) and with the mixed cluster expansion (35.47),

i b;(l) Gn(l, ... , n)= - il (I) Gn(1, ... , n) + Gn + l (l, 1, ... , n)

n+l

= L Sh(l, 1X1' •.. , lXi _ l ) Gn_i+l(lXi, ... , IXn) i=2

n~l bh_l(1X1, •.. ,lXi _ 1) , = ,L- SGn _ i +1 (lXi' .•. , IXn) bf (1') i2(1,1). 1=2. 1

(37.7)

From this we conclude

b n+l b

bil (I) Gn(1,···, n) = i~2 SGn_i + 1 (1X1' ... , IXn_i+ 1) bil (l) h-l (IXn_i+ 2' ... , IXn),

{IX;} =(1, ... , n). (37.8)

Here, the summation S accounts for the (, n ) different combinations of the 1-1

variables 1 to n. When (37.8) is inserted into (37.5) this summation can be carried out because of the symmetry of the interaction coefficients:

b bil (I) Vt"'l(l, ... , k)

= I ~ Vl~~(l, ... , k, 1', ... , m') l:fb(l) im(1', ... , m') m=l m. u 1

(37.9)

= vl~l (1, ... , k, 1) + .... (37.9a)

If the cluster expansion is restricted to the four-point function, as in the selfenergy (36.91), the general formula (37.9) reduces to

b 3 1 b bil(l) v,.[4l (1, ... , k)~ m~l m! Vl!lm(l, ... , k, 1', ... , m') bil(l) im(1: ... , m'). (37.10)

This equation is drastically simplified if only cubic and quartic anharmonicity is assumed. In this case, we obtain expressions which are analogous to (36.68)

b VJnl (1,2)

bil (3)

b VJnl (1,2, 3)

bil (4)

b vin] (1,2, 3, 4) 0

bil (5) (37.11)


These expressions may be used to derive simple approximations of the vertex part integral equations.

Differentiating the self-energy expression (36.91) and using (37.10), we see that all instantaneous interaction coefficients are of the type v;,[41. This kind of renormalization is important in the case of highly anharmonic crystals (GbTzE and MICHEL, 1969; BECK and MEIER, 1970). For simplicity, we drop the superscript [4J in the following. The interaction coefficients have to be understood, however, as renormalized. From the first two terms of (36.91) we derive, among other terms, the approximate relation

r3 (1,2,3) = V3 (1,2,3) - i [!- V4 (1, 2, 5,6)

- i V3 (1,4', 5) f2 (4',4") V3 (2,4'; 6)J f2 (5,5') f2 (6,6') r3 (5', 6', 3). (37.12)

This equation is shown in diagrammatic form in Fig. 37.2a. It involves two important approximations. Firstly, it is a linearized equation insofar as some of the cubic vertex parts have been replaced by the instantaneous interaction vertex V3 • Secondly, the right-hand side is only symmetric in arguments 1 and 2. Complete symmetry in all three arguments 1, 2 and 3 has been abandoned. This approximation, however, still enables symmetric self-energy expressions to be derived when r3 is inserted into (37.1). The iteration of (37.12) leads to ladder diagrams and chain diagrams.

Within the approximation, defined by (37.12), the analogous equation for the quartic vertex part follows by differentiation

r 4 (1 , 2,3, 4) =rp)(I, 2, 3,4)- i [t V4 (1, 2, 5, 6)

- i V3 (1,4', 5) f2 (4',4") V3 (2,4'; 6)] f2 (5,5') f2 (6,6') r4(5', 6', 3,4). (37.13)

a~=t>+C»+D

b ~= [Q]+ CD + [)JI

-f 2

Fig. 37.2a- c. Diagrammatic representation of vertex part integral equations (a and b), which provide the summation of mixed ladder and chain diagrams as, for instance, b. The renormalized

instantaneous interaction vertices are indicated bv ODen trian!!les and sauares


Here, the inhomogeneous term

rp)(I, 2, 3, 4) = V4 (1, 2, 3, 4) - [1 V4 (1, 2, 5, 6)

- i V3 (1,4', 5) f2 (4',4") V3 (2,4'; 6)]

. G~F(5, 6; 5', 6') V3(5; 7, 3)j~(7, 7') V3(6; 7', 4) (37.14)

is no longer an instantaneous quartic interaction vertex but depends on the time-ordered four-time Green function G 4 in the Hartree-F ock approximation (37.4). The Eq. (37.13) is represented by the diagrams in Fig. 37.2b.

The integral equations (37.12) and (37.13) have identical kernels. Their solutions can therefore be calculated from the corresponding Green function, here denoted by D, which is the solution of the same kind of integral equation but with a unit inhomogeneity

D(1,2; 3,4)=bA1A30"13b(tl-t3)bA2A40"24b(t2-t4)

- i K (1,2; 5,6) f2 (5,5') f2 (6,6') D(5', 6'; 3,4), (37.15)

K (1,2; 5,6) =1 V4(1, 2, 5,6) - i V3 (1,4', 5) f2 (4', 4") V3 (2,4'; 6). (37.16)

If the Green function D is known, the cubic and quartic vertex parts can be represented as

r3 (1,2,3) = D(I, 2; 1', 2') V3 (1',2', 3),

r4(1, 2, 3,4) =D(1, 2; 1',2') rp)(1', 2', 3,4).

(37.17)

(37.18)

The equations (37.1) to (37.4) and (37.15) to (37.18) constitute an approximation by which two-phonon correlations can be treated in a systematic manner in the Green functions G2 , G3 and G4 • Dynamical effects of pairs of phonons can be studied by these relations, which include the pair-pair interaction K of (37.16) and the pair-single phonon interaction Vj41.

In this section, all time arguments vary along the full path L. When performing actual calculations, one has to distinguish the different ways in which the time arguments can be distributed over the partial paths L + and L _. This may be done by writing the Green function D and the other four-time quantities as matrices of rank four. Defining the order of the elements by the array

[

( + +, + + ) (+ +, + - ) ( + +, - +) (+ +, - - )J (+-, ++) (+-, +-) (+-, -+) (+-, --) (-+, ++) (-+, +-) (-+, -+) (-+, --) ,

(--, ++) (--, +-) (--, -+) (--, --)

(37.19)

we write, for example, the inhomogeneous term of the integral equation (37.15) in the form

~J o . 1

(37.20)

Similarly, the pair-pair interaction K is given by the matrix


The mu1tiplation of such matrices involves double time integration from -;- 00 to + 00 with suitably chosen signs.

In the physical limit r = r =-j, the eight components r3 (1 ±, 2 ±, 3 ±) of the cubic vertex part are not independent. They obey a relation which is similar to (36.71). The latter is derived by varying j+ and r simultaneously in the relation

(37.22)

This defines a combined functional derivative with respect to the umque physical field, which can be expressed as

(37.23)

Doing this on both sides of (37.22) one arrives at (36.71). By continued differentiation similar relations can be found for the higher n-point functions. By the definition (36.34), we have

vl oo ] (1 +) = Vl oo ](1_), U+ = r). (37.24)

Analogously we are varying here /1 (1 +) =/1 (L) =/1 (1) with

(37.25)

This leads to the relation

1:(1 +,2+)-1:(1 +,2_) = 1:(1_,2+) -1:(L, 2_). (37.26)

Repeating the differentiation, we obtain

r 3(1+, 2+, 3+)-r3(1+, 2+, 3_)-r3(1+, 2_, 3+)+r3(1+, 2_, 3_)

=r3(1_, 2+, 3+)-r3(1_, 2+, 3_)-r3(1_, 2_, 3+)+r3(1_, 2_, 3_). (37.27)

Similar relations hold, obviously, for the higher-order vertex parts. In many problems, which one can treat with the general expressions (37.1) to

(37.3) it is sufficient to calculate D(1, 2; 3,4) under the special condition that t3 and t4 are equal and on the same path. This is obviously the case in (37.17). The expression (37.18) for r4 depends on D as a general four-time function. However,


in cases where r4 is used in (37.3) and where G 4 IS multiplied with an instantaneous, symmetric interaction vertex

G 4(1,2,3,4) Vn(3, 4, ... ),

the second and the last two terms of (37.3) give rise to

2/2 (1, I') 12 (2,2') D(I', 2'; 3,4) y"(3, 4, ... ).

(37.28)

(37.29)

Here, also, the Green function D has only to be known as a three-time function. One may, therefore, reduce D to a 4 x 2 matrix, where, in the scheme (37.19), only the first and last columns are taken.

The expression (37.29) can be used to evaluate the time-ordered double-time Green function

G~(M I N)= (TM(t) N(O), (37.30)

where M and N are given as series expansions in terms of phonon normal coordinates with instantaneous coefficients

M(t) =M I, t(l) A(I) +~ M 2, t(l, 2) A(I) A (2),

N(0)=N1 ,t(2) A(2)+~N2, 0(3,4) A(3) A(4). (37.31)

Here, the coefficients differ from the definition (36.11) by an additional delta function

(37.32)

Such expressions arise in calculating second-order infrared and Raman spectra. Inserting them into (37.30) we obtain, with (37.2), (37.3), (37.17) and (37.29),

G~(M I N) = M I, t(1) 12 (1,2) N1 , 0(2)

+iM 2,t(l, 2)/2(1, 2)/2(3, 4) N2, 0(3, 4)

+ ~ M 2, t(l, 2) 12 (1, I') 12 (2,2') D(I', 2'; 3,4) N2, 0(3, 4). (37.33)

Here, the first term represents the first-order spectrum with renormalized firstorder coupling coefficients

M I, t(1) = M I, t(l)

-~ M (I' 2') 1 (I' 1/1) 1 (2' 2/1) D(I/1 2/1· 3' 4') V (3' 4' 1) (37.34) 2 2, t' 2' 2, '" 3"

and, similarly,

N 1 , 0(2) =Nl,O(2)

-~ N2, 0(3, 4)/2(3, 3')/2(4, 4') D(3', 4'; 3'; 4/1) V3 (3'; 4'; 2). (37.35)

The second terms of these expressions and the last term of (12.33) have the same structure as the corrected bubble diagram contribution to the self-energy. The Green function of (37.33) can therefore be calculated after a detailed analysis of the vertex corrections in the self-energy expression (37.1). The second term in (37.33) is a constant, which drops out in calculating the retarded Green function G;(M I N). The third term represents, with respect to the external interactions, the pure second-order spectrum.


b) Self-energy with vertex corrections. Starting from the general relations and expressions of the preceding Sects. 36d and 37 a, one can develop a variety of approximations extending the renormalized harmonic approximation. To account for finite phonon widths it is necessary to consider in the self-energy at least the expression (37.1), i.e. the diagram in Fig. 37.1a. In most cases, it is sufficient to treat this quantity without vertex corrections, i.e. with r3 replaced by the instantaneous interaction coefficient V3 • In this approximation, one obtains the so-called bubble diagram contribution to the self-energy (Fig. 37.3a), which is discussed in Sect. 38a. The elaboration of the complete expression (37.1) is important, for example, for the detailed theory of the effects of anharmonicity on the sound propagation in a lattice. In particular, the difference between the isothermal and adiabatic elastic constants and the transition from the lowfrequency sound regime to the high-frequency acoustic phonons, which are measured in Brillouin and neutron spectroscopy, can be treated satisfactorily only by calculating the cubic vertex part.

From (37.1) we derive the retarded self-energy contribution, dropping the superscript [4],

niAk(t)=I(i+, k+)-I(i+, k_) (tj=t, tk=O)

i 00 00

= -:2 L V3 (A j, ..1.1' Az) S dtl S dtz Al.h -00-00

. {f:'(t-tl)fA~(t-tz) rA!A;dtu t z , 0)

-f:,(t-t1) 1;.,(t-tz)* rt-;nk(t1, t z , 0) 1 2

(37.36)

Here, the two-point function has been treated for the sake of simplicity in the 'diagonal' approximation, which is defined in detail in Sect. 38a (Eqs. (38.2) to (38.7)). In this approximation the spectral representations of (38.8) and (34.79) can be used to study expressions like (37.36) in a general manner. In (37.36) the cubic vertex part occurs in four combinations,

++ --I;fA;Ak(t1, t z , 0)=1;(..1.1' t1±; Az , tz±; Ab 0+)

-1;(I1,t1±;Iz ,tz ±;Ak,0_), (37.37)

which differ only with respect to the distribution of the two time arguments t 1

and t z over the paths L+ and L_. Equation (37.36) is further reduced by using the relation (37.27) and by introducing the combinations

The Fourier transform of (37.36) is then obtained as

(37.38)

(37.39)


(37.40)

The double Fourier transforms of the vertex part components are defined as 00 00

I";.*A*A (Wl' W2)= S dt l eiW1tl S dt2 eiW2t2 I";.*A*Ak(tU t 2, 0). 12k 1 2

(37.41) -00 -00

The third term of (37.40) can be combined with the first two terms by inserting the spectral representation of (34.79) and by performing the w'-integration. To carry out the integration the component r+ + is divided into two part, P and ra ,

(37.42)

which are analytic with respect to w' in the lower and upper halves of the complex w'-plane, respectively. The self-energy contribution, obtained in this way, may be written in the following two forms

1 OCJ 00

IIA'Ak(w)=-t L V3(Ai,Al ,A2)Z S dxlmG~l(x) S dylmG~2(Y) A1 A2 n - OCJ -00

1 a -------:.- {(1 +n(x)) I";.*,\*,\ (-x, w+x) x+ Y+W+lI; 12k

+(1 + n(y)) r/',VAk(w +y, -y) 1 2

-nA*A (-X, W+X)-I";.~A*)JW+ y, -y)} (37.43) 1\.1 2 k 1 2

=-t L V3(Ai,Al,A2)~ S dW'{ImG~l (~+W')G~2 (~-w,) A1A2 n -00

. [n (~-w,) r4 AiAk (~+w" ~-w,) +I";.rA;Ak (~+w" ~-w,) n. (37.43 a)

With these expressions the corrected bubble diagram contribution can be evaluated for any approximation of the cubic vertex part.


a < => e <C)C>

b E => f <X> c ~ ~ d ~ 9

k Fig. 37.3. Low-order dispersive contributions to the self-energy IT Ai))W), Symbols as in Fig. 37.2

Within the ladder and chain diagram approximation developed in Sect. 37 a, one obtains with (37.17) a low-order approximation of the cubic vertex part from the first iteration of the integral equation (37.15)

D(I)(1, 2; 3, 4) = b Al A3 () 13 ,,(t 1 - t 3) "A2A4 () 24" (t 2 - t4 )

-iK(1, 2; 3',4')12(3',3)12(4',4). (37.44)

Inserting this expression with (37.16) and (37.17) into (37.43) one finds three different self-energy contributions which are represented by the diagrams a, e, and f of Fig. 37.3. The first term, diagram a, which arises from the delta functions in (37.44), is the well-known bubble diagram contribution, which is discussed in Sect. 38a, (38.15). The nontrivial term in (37.44) leads to two contributions corresponding to diagrams e and f, respectively. The first of these two is given by

niA.(w)e =t L L V3(A i , ..11' ..12) G~,A1A2(W) V4(Ai, kL ..13, ..14) A1A2 A3A4

. G~,A3A.(W) V3(A~, A:, Ak ), (37.45)

with the retarded two-phonon Green function (cf. (38.18))

1 00 00 1_e-Pn (x+ y) G~ A A (W) =2' S dx 1m G~ (x) S dy 1m G~ (y) . (37.46)

, 1 2 11; -00 1 -00 2 X+Y+W+IB

The diagram f is the simplest ladder diagram having only one ring. It IS

related to the expression

llAiAk(W)f= - L L L V3(Ai , ..11, ..12) V3(Ai, ..10, ..13) V3(Ai, ..16, ..14) V3(A~, A:, Ak) A1A2 AO A3A4

100 00 00

• 11;3 S dx S du S dvlmG~l(x)GL(w+x)ImG~o(u) -00 -00 -00

. [(1 + nJ (1 + nJ (1 + nv) - nx nu nv]

.{ 1 ImG~(v)G~4(w-v)+ 1 . G~(w+X)ImG~(V)}, X+U+V 3 X+U+V+W+IB 3 4

nx = n(x), etc. (37.47)

Here, the index 0 denotes the exchanged ring phonon.

Sect. 37 Vertex renonnalization 519

From the viewpoint of perturbation theory it is of interest to consider also the diagrams c and d of Fig. 37.3 which contribute to the same order V32 V4 as diagram e. The self-energy contribution corresponding to diagram d is obtained from (37.43) but the vertex part has to be calculated in an approximation different from that in (37.12). Differentiating the self-energy of (36.91) one obtains, besides the right-hand side of (37.12), the following terms

13(1,2,3) = ... -~ V141(1, 3, 2', 3') 12(2',2") 12(3',3") 13(2",3",2)

-~ V141(1, 2', 3') 12(2',2") 12(3', 3") ~(2", 3", 2,3)+ .... (37.48)

If, here, one takes for the vertex parts r.. the renormalized instantaneous interaction vertices V!41, both terms contribute equally to the self-energy (37.43). Their sum leads to

(37.49)

Besides this contribution to the self-energy there exists a similar one which corresponds to the related diagram c of Fig. 37.3. It follows from the last term of (36.91), replacing the vertex parts 13 by the renormalized instantaneous interaction coefficients V141, and is given by

lIA'AJw)c=! L L V4(Ai , Al , A2 , A3) V3(At, A~, A4) V3(A~, A~, Ak) A,A2 A3A4

1 00 00 00

· 3 J dx J dy J dv ImG~,(x) ImG~2(y) n -00 -00 -00

· [(1 +nx ) (1 +ny) (1 +nv)-nx nynvJ

· { 1 . ImG~ (v) G~ (w+v) x+Y+V+W+IB 3 4

+p 1 G~3(W+V) ImG~4(V)}. x+y+v

(37.50)

Finally, it should be mentioned that a self-energy term corresponding to diagram g of Fig. 37.3 does not occur in the expansion of the self-energy in terms of the exact two-point function. This diagram is meaningful if the internal lines denote the propagation of bare harmonic phonons. The corresponding contribution to the self-energy is, however, already included in the expression (38.15) for the bubble diagram a.


To perform the summation of the complete set of ladder and chain diagrams in the self-energy and in (37.33) to (37.35), it is necessary to calculate the vertex part and the Green function D on the basis of the full integral equation (37.15). In particular, it is necessary to calculate Green function components D± ± which are defined by relations analogous to (37.37)

DiJ2,).3).4(t U tz, 0)=D(,,1, 1 , t 1 ±; ,,1,ztz±; ,,1,3,0+; ,,1,40+) -D("1,l,tl±;,,1,ztz±;,,1,30_,,,1,40_). (37.51)

These quantities are not independent but obey, as a consequence of (37.27), the relation

(37.52)

It is, therefore, only necessary to calculate from (37.15) the three components D + +, D + -, and D - +. Taking the double Fourier transform, inserting spectral representations and performing contour integrations, one obtains for the first one

1 00 { .- S du [G~ (u-w')+2in(u-w') ImG~ (u-w')J n 0 0 -00

. F).~).~, ~3).4 (I+ U' I- U )

+[1+n(u-w')] ImG~o(u-w')G~5 (I+ U )* G~6 (I- u)

.DX~).~').3).4 (I+u'I- U )

+n(u-w') ImG~o(u-w') G~5 (I+ u) G~6 (I- u)*

. D~~).~, ).3).4 (I+ U' I- u)} (37.53)

with

(37.54)

and

F).~).~').3).4 (I+ U' I- U ) =ImG~5 (I+ U ) G~6 (I- U )

. [n (I+ U ) D~~).P3).4 (I+ U' I- U ) +D~~).~').3).4 (I+ U' I- U)]


+G~5 (I+ U) ImG~6 (~ -u)

. [n (~ -u) D1pP3 ).4 (I+u, I- u) +D~PP3).4 (~ +u, ~ -u) l (37.55)

On the right-hand sides of these equations, the Green function components DX

(x = c(, /3, y, b) are defined, by analogy with the relations (37.38), (37.39) and (37.42), as

D"=D++-D-+ (37.56)

DP=D++ -D+- (37.57) and

DY+D~=D++. (37.58)

The subdivision of D++ (~ + co', ~-co') into the two parts DY and D~, which are

analytic with respect to co' in the lower and upper halves of the complex co'plane, is not unique. In the following we include the regular co'-independent part arbitrarily in DY and write

(37.59)

and

Dt).2,.l.3).4 (~ + co', ~-co') = L L V3 (A 1 , Ao, As) V3 (A 2 , A~, A6 ) .1.0 ).5).6

'{--;JdUJdV, 1 . n(U)ImG~0(U)F).').').).4(C02+v,co2-V) 11: -00 -00 co +U-V+16 5 6' 3

+~ -L dun(u) ImG~O(u)G~5 (I+co'+u) G~6 (~ -co'-u)*

(37.60)

The integral equations for D" and DP are obtained by deriving first from (37.15) the equations for D-+ and D+-:

522

and

Dynamical theory of interacting phonon systems

D;,12,A3A4 (I+W', I-W') = I I V3 (A 1 , Ao, A5) V3 (A z, A6, A6 ) AD A5A6

. ~ -I du n(u-w') 1m G~o(u-w') {2iFA;A~' A3A4 (I+U' I-U)

+G~5 (I+ur G~6 (I-U) [-D~~A~'A3A4 (I+U'I-U)

+D~PP3A4 (~ +u, I-U)]

+G~5 (I+U) G~6 (I-U r D~PP3A4 (I+U' I-u)},

Di,l2,A3A4 (I+W', I-W') = I I V3 (A 1 , AO' A5) V3 (A z, A6, A6 ) AD A5A6

Sect. 37

(37.61)

. ~ -I du[l+n(u-w')] ImG~o(U-W'){2iFA;AP3A4 (I+U'I-U)

+G~5 (I+U) G~6 (I-ur [-D1~A~'A3A4 (I+U'I-U)

+D~~AP3A4 (I+U' ~ -u)]

+G~5 (I+U r G~6 (I-U) DIp~'A3A4 (I+U' I-u)}. (37.62)

They lead, with the Eqs. (37.53) to (37.57), to the integral equations

and

. [n(U)D~~A~'A3A4 (I+ w' +u, I-W'-u)

+D~;AP3A4 (I+W' +u, I-W'-u) n, (37.63)


. [(1+n(U))D~V'~'A3A4 (~ +CO'+U'I-CO'-U)

-D1;A~'A3AO (~ +CO'+u,I-co'-U)]). (37.64)

From these equations it can be seen that D" and DP are analytic with respect to co' in the lower and upper halves, respectively, of the complex co'-plane. All four components DX and the function F are analytic with respect to co in the upper half-plane, as one expects for retarded quantities.

The equations (37.59), (37.60), (37.63) and (37.64) constitute a set of four coupled integral equations, by which the Green function D is completely reduced to the familiar retarded one-phonon Green function. These equations represent a general basis for studying collective effects of phonon systems, e.g. the transport properties. Phonon transport equations can be obtained from the relations presented here in a manner which is closely related to the derivations given by SHAM (1967) and KLEIN and WEHNER (1968, 1969).

The self-energy can be calculated in terms of the four functions DX using the relation

qA;Ak (I+ co', ~ -co') = 21. D~PP3A4 (~ + co', ~ -co') V3(A~, A:, Ak ),

(x=rx, p, y, b). (37.65)

Inserting this into (37.43a) and using the abbreviation of (37.55), one can write the corrected bubble diagram contribution in the form

IIAiAk(co)corr. bubble = -t L L V3(A i , A1 , A2 ) AI A2 A3AO

. ~ -I dco' FAPP3A4 (~ + co', I-CO') V3(A~, A:, Ak): (37.66)

This self-energy expression allows studies to be made of the coupling of collective excitations, such as second sound or two-phonon bound states, to the elementary phonon excitations.


38. Simple approximations and results

a) Self-energy and retarded Green functions. In many cases, particularly for perturbation-theory treatments, it is sufficient to calculate the anharmonic interactions in simple approximations. In this section a few results will be derived from the general relations developed so far. In order to do this it is necessary to carry out explicitly the time integrations along the double path L =L+ +L_ introduced in Sect. 35a. From the simple examples given in the following it will become clear how the more complicated approximations discussed in Sect. 37b have been worked out.

The basic restriction in this section is to take the vertex parts in their lowestorder approximations, r,,(O)(1, ... , n) = y"(1, ... , n), i.e. to neglect all of the socalled vertex corrections. Doing this, we disregard also the instantaneous renormalization of V3 as given by (36.68). In this way we obtain from (36.91), without the last term,

l'(1, 2) =l'[21(1, 2) -~ V3 (1,3,4) f2 (3,3') f2(4, 4') V3(3', 4', 2)

-~ V4 (1, 3, 4, 5)f2(3, 3') f2 (4, 4')f2(5, 5') V4 (3', 4', 5', 2). (38.1)

To simplify this further, we treat the two-point function as diagonal with respect to the phonon indices

rf 2(1+,2+) f2(1+, 2_)] = b- [f(,(t 1-t2) JA,(t2-t1)] (38.2) li2(L, 2+) f2(L,2_) (J A,A2 JA,(t1 -t2) fA~(t1 -t2) .

The functions on the right-hand side, according to (35.26) to (35.29) and with (35.36), are identified as

f/ (t) = (T> A(.?c, t) A + (.?c, 0) - (A (.?c) 2

fA< (t)= (T< A(.?c, t) A + (.?c, 0) - (A (.?c) 2

JA(t) = (A(.?c, t) A + (.?c, 0) - (A(.?c)2. (38.3)

Here, J denotes the dynamical part of the correlation function; the constant C2

of (36.61) is assumed to vanish, so that we can write

1 00

f/(t=O)=ft(t=O)=JA(t=O)=- J dw[1 +n(w)J ImGHw). (38.4) 1t -00 .

In the harmonic approximation of the Hamiltonian (36.3) these quantities are equal to 2nA + 1.

The four functions of (38.2) obey the relation (36.71)

fA> (t)-JA(t)-JA( -t) + ft (t)=O. (38.5)

Using this relation and (36.70), we obtain

f/(t)=G1(t)- (A (.?c) 2 = -i GHt)+JA( -t),

ft (t) = - f/ (t) + JA(t) +JA( - t)=i G~(t) +JA(t). (38.6)

Furthermore, a few simple properties are easily verified

Sect. 38 Simple approximations and results 525

fA~(-t)=fA~(t), JA(-t)=JA(t)*, ft(t)=f/(t)*. (38.7)

The Fourier transforms of the time-dependent functions in (38.6) can be expressed by the spectral representations, (34.78) to (34.80), with the result that

~ i 00 1 1 fA'<(w) = ±-2 S du ( . _. ) JA(u), ft(w) = f/(w)*. (38.8)

n -00 W-U±lB W+U+IB

The retarded self-energy, following from (38.1) according to (36.73), is written as

(38.9)

The instantaneous part .r[2], which is given by (36.44) and corresponds to the diagrams a to d of Fig. 36.2, leads, with (38.7), to

II~~]A2 = V2(A I, A2)+ L V3 (A I, A2, A3)<A(A3 )

A,

+tI, V4(A I, A2, A3 , Aj)(2nA, + 1). (38.10) A,

The thermal expectation values <A(A) may here be taken from (36.31). Dispersive interactions of the simplest type involve three-phonon processes,

as represented by the second term of (38.1). The corresponding diagram is the so-called bubble (diagram e of Fig.36.2 with bare interaction vertex). Its contribution to the retarded self-energy is obtained by defining first, according to (36.73), the following retarded Green function

i {f2 (AI t+, A~ 0 +) f 2(A 2, t+; A~, 0 +) - f2(A I, t+; A~, 0 _) f2 (A2' t+; A~, O_)}

=:()llA\ ()l2A'z G~,A1A2(t). (38.11)

This is the two-phonon Green function in the Hartree approximation, as becomes clearer by writing

G~,A,;.,(t)=i {<TA(A I, t) A +(AI' 0) <TA(A2' t) A +(A2' 0)

- <A +(AI' 0) A(AI' t) <A +(A2' 0) A(A2' t)}

=: i {<TA(A I, t) A(A2' t) A +(AI' 0) A + (A2' O)H

- <A +(AI' 0) A +(A2' 0) A(AI' t) A(A2' t)H}

=: G~' H(A(AI) A(A2) I A +(AI) A + (A 2))·

(38.12)

(38.13)

(38.14)

Using (38.11), one finds for the bubble diagram contribution to the retarded selfenergy the expression

IIA1A,(w). = -t L V3 (A I, A3 , A4) G~,A,A.(W) V3 (Aj, A!, A2), (38.15)

where A,A.

00

G~'A1A2(w)=i S dteiwt {fA~(t) f (,(t)-JA,( - t) JA2 ( - t)}, (38.16) -00

is, in shorter notation, the Fourier transform of (38.12). In this approximation the retarded two-phonon Green function is calculated by introducing the


Fourier transform 1 ()()S . ,

f/(t) =-2 dw' e-"" tf;.> (w'), n _()()

(38.17)

and using the spectral representation of (38.8). After contour integration one obtains

1 ()() ()() 2(u+v) G~'A1A2(W)=(2n)2 S du S dv ( ? ( .? JA1 (U)JA2 (V). (38.18)

-00 _()() U+v - W+18

In the case of harmonic phonons, the remaining integrations can be carried out by inserting the spectral density function of (34.48)

r.(O) _ 2(Wl +(2) G2 A A2(w)-(1 +n1 +n2) ( )2 ( ')2

, 1 WI +W2 - W+18

2(Wl -(2) +(n2 -n1) ( ? ( . )2' WI -W2 - W+18

(38.19)

In this approximation the retarded two-phonon propagator is given by terms which are of the same form as the harmonic one-phonon propagator in (34.50). The resonance frequencies, however, are given by the sum and difference of two phonon frequencies. In the self-energy expression (38.15) the various terms of G~O) describe decay and scattering processes.

The last term in (38.1) (diagram g of Fig. 36.2 with bare interaction vertex) is treated in the same way as the foregoing one. The result accounts for fourphonon processes in the form

IIA1A2 (w)g= -f, I V4 (A 1, A3, A4 , As) G~,A3A4A5(W) V4 (Aj, A:, A~, A2) (38.20) A3 A4A5

with the three-phonon Green function in the Hartree approximation,

()()

G~,A1A2A3(w)=i S dteiwt{fA~(t)fA~(t)fA~(t) -JA,( -t)JA2 ( -t)JA3 ( -t)} (38.21) -()()

In the harmonic approximation the three-phonon Green function is given by

r(O) 2(W1+W2+W3) Gi,A1A2A3(w)=(1+n1 +n2+n3+nln2+n2n3+n3nl)( )2 ( ')2

WI +W2 +W3 - W+18

{ 2(Wl +W2 -(3) } + [(1+nl+n2)n3-nln2J 2 ( . z+cycl.perm ..

(WI +W2 -(3) - W+18)

(38.23)

The expression (38.18) and (38.22) for the retarded two- and three-phonon Green functions are easily extended to higher phonon numbers.

The self-energy expressions derived in this section rest on the assumption that the two-point functions are diagonal. This assumption can always be fulfilled within a harmonic approximation. The interactions, however, generally


lead to a nondiagonal self-energy matrix and to a coupling of the phonon modes in the Dyson equation. In the case of a perfect lattice only phonons with the same wave vector can be coupled. If there is only one branch, the Dyson equation (36.75) reduces to the simple equation

(38.24) with

IIJJw):= IIJ..*J..(w),

= L1 J.. (w) - iT;, (w). (38.25)

The solution of (38.24) is given by

2w Gr (w) = )..

).. w~-(w+iB)2+2w)..IIJ..(w)' (38.26)

The poles of this Green function define the frequency cO).. and width 2T;, of the renormalized phonon A. These quantities are determined from the vanishing of the denominator. If the damping is weak, so that we have a well-defined resonance, the condition is

(38.27)

With the renormalized phonon frequencies cO).., and neglecting linewidths, a further approximation can be defined which is sometimes called pseudoharmonic approximation (see Sect. 3d). The condition for self-consistency, cO).. =w).., is simply expressed as

(38.28)

The half-width of the renormalized phonon follows as T;, :=T;,(cO)..). If, in the case of self-consistency, one puts

(38.29)

the Green function (38.26) describes a Lorentzian lineshape

(38.30)

Generally, a finite imaginary part of the self-energy is related to nontrivial forms of the spectral density function

(38.31)

This general expression is important, for example, for including renormalization effects in the Green functions of (38.18) and (38.22).

The coupling between single phonon modes is treated by solving the Dyson matrix equation which, in short notation, is written as

Gik = Gl?) bik - I Gl?) IIi! G1k • (38.32) 1


This may be done in two steps. Firstly, retaining only the diagonal elements of the self-energy, we define, analogously to the Green function (38.26),

G(l)- 1 ii [G~O)] -1 + II.·

n II

(38.33)

Eliminating with this equation the zero-order Green function in (38.32), we obtain a different Dyson equation

Gik=G~t)bik- I GU) IlilG1k · I

(l*i)

(38.34)

Here, only the off-diagonal elements of the self-energy matrix are left. This equation can be solved in simple terms if only two coupled modes are considered, (i, k, 1= 1, 2)

1 Gll =[G(1)]-1-Il G(1)Il

1 12 22 21

G12 = - GW Il12 G22 , etc.

(38.35)

(38.36)

The poles are in this case determined by the zeros of the denominator of G 11' i.e. from the equation

(38.37)

Neglecting the frequency dependence and the imaginary parts of the self-energy, we find the renormalized frequencies as

wi =1 [wi + w~ ±v(wi - W~)2 + 16w 1 w 2.d 12 .d 21 ],

wf =wf +2Wi .d U •

(38.38)

In addition to the frequency shifts, the mode-mode coupling leads generally to lineshape effects which can be treated by a full analysis of the Dyson equation.

Occasionally, it is of interest to calculate explicitly the Green functions G~(A(A1)A().2)IA(A3)) and G~(A(A1)IA(A2)A(A3)) which are mixtures of oneand two-phonon Green functions. They can be derived from the time-ordered three-time Green function

G3 (1,2, 3) = - i12 (1, 1') 12 (2, 2') 12 (3, 3') 1; (1',2', 3') 11 = 0. (38.39)

Neglecting vertex corrections, as is done throughout this section, we write this relation in the form

<TA(A1' t 1) A(A2' t2) A(A3' t3)

= - i I S dt'f2 (,11' t 1 ; A~,t') 12 (,12' t 2; A~,t') 12 (,13' t 3; A~, t') V3 (A~, A~, A~), (38.40) A1A2A3 L

where t 1 , t2, and t3 can be anywhere on the path L. Putting t1 = t2 = t and t3 = 0, all three on the path L+, we have

Sect. 38 Simple approximations and results

(TA(A 1 , t+) A(A2, t+) A(A3, 0+) == (T> A(A1 , t) A(A2, t) A(A3, 0)

== G~(A(Al) A(A2) I A(A3)) 00

= -i J dt' {fA~(t-t')fA~(t-t')fA:( -t') -00

529

-JA, (t' - t) J A2(t' - t) JA3 (t'n V3(Ai, A~, A~).

(38.41)

On the other hand, choosing tl =t2=t on the path L+, but t3=0 on the path L _, we obtain from (38.40)

(TA(A 1 , t+) A(A2, t+) A(A3, 0 _) == (A(A3' 0) A(A1 , t) A(A2, t)

==J _t(A(A3) I A(Al) A(A2)) 00

= -i J dt' {fA~(t-t')fA~(t-t')JA3( -t') -00

-JA,(t' -t)JA2(t'-t)fA~( -t')} V3(Ai, A~, A~).

(38.42)

By subtracting (38.42) from (38.41) we arrive at the retarded Green function. Taking the Fourier transform we find

G~(A(Al) A(A2) I A(A3)) = - G~'A'A2(w) V3(Ai, A~, A~) G~3(W). (38.43)

Similarly, one obtains

G~(A(Al) I A(A2) A(A3)) = - G~,(w) V3(Ai, A~, A~) G~,A2A3(w). (38.44)

These Green functions are used in calculations of first- and second-order infrared and Raman spectra. There one considers retarded Green functions G~(MIN) and spectral density functions Jw(MIN), where M and N are series expansions of lattice dipole moments or electronic polarizabilities in terms of phonon normal coordinates

M=LM1(A1)A(A1)+t L M2(Al,A2)A(Al)A(A2)' A, A,A2

N=LN1(A2)A(A2)+t L N2(A3,A4)A(A3)A(A4)' (38.45) 1.2 1.31.4

An approximate expression for the corresponding time-ordered Green function G~(MIN) is derived in Sect. 37a. Neglecting the vertex corrections in (37.33), i.e. replacing the Green function D by delta functions, we obtain

G~(MIN)= L Ml,A,(W)G~,dw)N1,J.2(W) 1.,1.2

+t L M2(Av A2) G~'A'A2(w) N2(Ai, A~). (38.46) 1.,1.2

Here, the second term represents two-phonon spectra due to direct coupling of an external field to pairs of phonons. The first-term is written as a renormalized one-phonon spectrum. The one-phonon Green function G~'A2 follows from the Dyson equation (38.32) where, with the self-energy expression (38.15), the


excitation of two-phonon spectra by cubic anharmonicity can be described. In addition, the renormalized first-order coupling coefficients

MI,).,(OJ)=M&{I)-t L M2(A~ A~) G~,).').2(OJ) V3(A'I*' A~*, AI), etc. (38.47) ;.,)., contain the two-phonon excitations in the form of interference terms.

Up to here, practically all simple relations concerning retarded quantities have been summarized. Any improved approximations and more complete treatments deal with high-order anharmonicities and require the anayls'is of the vertex-part functions.

b) Free energy. The static properties of a crystal in thermal equilibrium, being specified by temperature and internal and external static forces, are completely determined by the free energy of (35.32)

1 F= --logZ Z=Tre-PH.

f3 ' (38.48)

With the harmonic Hamiltonian of (34.21) the calculation is straightforward and

leads to {h 1 }' F(O) = ~ '2 OJ). - Ii 10g(1 + n;.) . (38.49)

In the same approximation the entropy is given by ~F(O)

S(O)= __ u_ oT

=k L {(1 +n).)log(l +n).)-n;.logn).}. ).

(38.50)

These expressions are characteristic for the noninteracting gas of thermally excited phonons.

In many cases we want to know the free energy as a function of a parameter 1" which appears in the Hamiltonian, H(1"). The calculation of F(1") can be facilitated by a simple and useful relation for the first derivative of the free energy with respect to 1"

of (1") 1 oZ(1") ----ar = - f3 Z (1") ----ar

=_1_ T { -PH«) OH(1")} Z(1") r e 01"

= /OH(1")) .. \ 01" <

(38.51)

To obtain the derivative in the latter form, one expands the exponential, differentiates by means of the Hellman-Feynman theorem, and orders the terms by cyclic permutation. The subscript 1" indicates that the thermal averaging is done with the 1"-dependent Hamiltonian. The above relation can be used to calculate the free energy of an anharmonic crystal, as treated by MARINCHUK

and MOSKALENKO (1963) and SHAM (1965). When (38.51) is applied to the


general lattice system, phonon normal coordinates may be used as defined above, either in the harmonic approximation or in any self-consistent approximation.

A simple formal way to treat phonon-phonon interactions is to multiply the interaction potential in (36.2) by a parameter 't", which is varied between zero and one

H('t")=H(O)+'t"V, O~'t"~1. (38.52)

Inserting this Hamiltonian into (38.51) we find

of('t") =(V) o't" t'

(38.53)

and, integrating, t

F('t")=F(O)+ S d't"' (V\" (38.54) o

F=F('t"=1).

Here, F(O) is given by (38.49) either with the harmonic frequencies or with selfconsistently calculated temperature-dependent frequencies. With 't" = 1 the second term is written explicitly as

1 00 1 1

S d't"(V\=h L, L v,,(.A·1··· An) S d't"(A(A1)···A(An)t· o n=1 n. A, ... An 0

(38.55)

By evaluating the integrals on the right-hand side, we can apply most of the arguments and relations discussed throughout this chapter in terms of 't"dependent quantities.

To determine the 't"-dependent equilibrium positions, we have to substitute for (36.22) the equation

11 (A 1)t = -~ {V1 (Ai) W;.,

00 1 } + m~2(m-1)! ;'2~;'m Vm(AtA2 ... Am) (A(A2)···A(Am)t . (38.56)

Here, the thermal expectation values on the right-hand side have to be broken up as above by the cluster expansion but with 't"-dependent n-point functions In(A 1···An)t·

A perturbation-theory expansion of the expectation values in (38.55) in powers of the interaction coefficients v" , i.e. in powers of 't", generally leads to terms of the form

1 1 S d't"'t"m-1 VmX=- vm X. o m

(38.57)

In this equation, X stands for 't"-independent products of propagators and expectation values. The quantities vm X are represented by diagrams with m vertices; their value divided by m yields their contribution to the free energy. A few terms of low order are listed in Table 38.1.

According to thermodynamical principles, the equilibrium configuration of a lattice at a particular temperature is related to a minimum value of the free


Table 38.1. Perturbation-theoretical contributions to the free energy

Diagram* LlF

I 1 a -hLIVI(AW~

A WA

b ! 2h L VI (Al)~ VI (A2)~ V2(A! Ail Al.h Wi W 2

C ~ -h L VI(AI)~ V3(AP2 A!)(2n2 + 1) A1).2 wl

d Q h - L V2 (U*)(2 nA + 1) 2 A

e 0 _~ L 1V2(AIA2)2e+nl+n2+ n2-nl ) 2A1A2 WI +W2 WI -W2

WI*W2 and W2--->WI for AI=A2

f 8 ~ L V4 (A I AP2 A!)(2n2 + 1)(2n2 + 1) 8 A1A2

g (}) _~ L 1V3(AIA2 A312 (1+nl +n2)(1+n3)+nln2+3 (1+nl +n2)n3-n1 n2) 6A1A2A3 WI +W2+W3 WI +W2-W3

* Symbols as in Figs. 36.1 and 36.2

energy. This means that the equations

of of1(A) =0 (38.58)

must be equivalent to the set of equations (36.22). Any approximate calculation of f1,! from (38.56) should therefore be in accordance with approximations used for evaluating the integrals in (38.55), so that the minimum conditions (38.58) are fulfilled; such approximations are called thermodynamically consistent. A similar thermodynamical requirement may be considered for the phonon frequencies. A self-consistent calculation of the frequencies from the equations of motion should result in best phonon frequencies in the sense that they minimize the free energy

(38.59)

Many approximate calculations of the free energy are described in the literature in connection with the variational techniques represented by (38.58) and (38.59)


that are used to determine the structure and the excitation spectra of crystal lattices.

In the following, the free energy is calculated in an approximation which is consistent with the renormalized harmonic approximation, as discussed in Sect. 36c. From (38.54) one obtains, integrating by parts,

F=F(O)+<V)- J drr O<V\. (38.60) o aT

Here, the first term is of the form of the right-hand side of (38.49). The phonon frequencies, however, have still to be considered as chosen by ansatz. Later, they will be varied in order to find the minimum free energy. The eigenvectors will not be varied here; for simplicity, they are assumed to be chosen in a way that ensures that the self-energy 1I~:! 1 is diagona1.

The second term on the right-hand side of (38.60) is the thermal expectation value of the effective interaction potential, which we write in the present approximation as

<V)[2]=h f ~ L Vn(A 1 ... A) <A(A1)···A(An)[2] n=l n. AI ••. An

=h m~o2m1m! VJ~(A1Af. .. AmA!)(2n1 +1) ... (2nm+1). (38.61)

Here, analogously to (36.32) and (36.40), the interaction coefficients are given by

VJ~(A1 Ai··· Am A!) = V2m(A 1 Ai ... Am A!)

+ f (l-~ )' L V;(A 1 Ai···AmA!A2m+1···AI)f1(A2m+1)··.j1(Al)·(38.62) 1=2m+1 m . A2m+I •.. AI

The equation (38.61) follows immediately from the definition of the cluster expansion, (35.42).

The T-dependent potential expectation value in the third term of (38.60) is consistently written as

<V)~2]=h f _1_ L VJ~(A1A~ ... AmA~)J2(A1A'1), ... f2(AmA~),. (38.63) m=O 2mm! AI ••. Am

Al ... l :n

The derivative of this quantity with respect to f1 (A)"

0<V)~2]

Ofl (A),

can be used to write (38.56) within the present approximation

2T 0<V)~2] f1 (A)t= -0" hWA Of1(A), . (38.65)

This relation serves to evaluate the third term in (38.60). Disregarding the derivatives of the two-point functions f2" with respect to T, we arrive at


1 a<V)[21 Sdn a ' o T

L J dn a<V)~21 af1(A), A 0 af1 (A), aT

= -(J ~2 L W A J dTf1(X) af a1 (A), A 0 T

h -= - (J"4 ~ W A f1 (A)f1 (A), (T = 1). (38.66)

Combining all three contributions we obtain the free energy in the following form

F =F(O) + (J ~ L W A f1 (A) f1 (,X) 4 A

+ h f _1_ L Vi~(A1 Ai ... Am A!)(2n1 + 1) ... (2nm + 1). (38.67) m=O 2mm! At ... Am

This expression represents the free energy in the renormalized harmonic approximation, provided the frequencies and eigenvectors are chosen in such a manner that the self-consistency conditions (36.56) are fulfilled. To see whether this is compatible with the thermodynamic requirements of (38.58) and (38.59), it is necessary to differentiate the free energy.

From the foregoing discussion it follows that the thermodynamic condition for the equilibrium positions,

aF h -0= af1 (A) =(J 2: wd1 (A)

+h f 2}m l L Vi~(U1Ai.··AmA!)(2n1+1) ... (2nm+l) (38.68) m=O • At ... Am

is identical to (36.58), which is the result of the dynamical theory. It can also be written as <1>l21(A) = o.

Differentiating the free energy with respect to the phonon frequencies, we find

(38.69)

The first term vanishes when the condition (38.68) for the equilibrium positions is fulfilled. Comparing the second term with (36.47) and (36.55), one concludes that the self-consistency conditions must be fulfilled to yield a minimum of the free energy expression (38.67).

The free energy of (38.67) can be rewritten in terms of the original lattice potential. Using the relations (36.8) and the self-consistency conditions, one can eliminate the second term and the m = 1 term from (38.67). The result is given by

-h f (;-:n1/ L <1>~~(A1Ai···AmA!)(2n1+1) ... (2nm+l). (38.70) m=2 . At ... Am

Sect. 39 Tables of elastic and dielectric constants 535

Here, the second term is the lattice potential corresponding to the equilibrium configuration. The third term represents anharmonic phonon-phonon interactions; the coupling coefficients cp~l~ correspond to the derivatives of the lattice potential at the equilibrium positions.

The third term in (38.70) demonstrates that in the renormalized harmonic approximation the free energy is no longer a sum of free energies of single quasifree oscillators as in (38.49). However, on differentiating the free energy with respect to temperature, one again obtains the entropy in the form of (38.50).

If the lattice potential had initially been expanded about the equilibrium positions as given by (38.68), we would by definition have fl(A}:::O. In this case, the lowest-order interaction term in (38.70) arises from quartic anharmonicity

LlF=-~ L CP4(A1Ap2A~)(2nl+1)(2n2+1). A, .1.2

(38.71)

Ordinary perturbation theory, starting from the harmonic approximation, leads within the same order to item f in Table 38.1, i.e. the same expression but with opposite sign. There is, however, no contradiction, because the zero-order freeparticle expression F(O) is calculated in these two cases from different phonon frequencies. The difference between the harmonic and renormalized harmonic frequencies is given in lowest order by (Eq. (38.10))

W~HA~W~A+tL V4(AA* Al Ai)(2n1 + 1). (38.72) A,

Inserting the renormalized harmonic frequency into (38.49) and expanding to first order in the frequency shift, we find

Fica.A =F~~ +~ L V4(A 1 Ai A2 A~)(2nl + 1)(2n2 + 1), A, .1.2

(38.73)

or twice the contribution of ordinary perturbation theory (COWLEY, 1963). Adding (38.71) and (38.73) yields the result of ordinary perturbation theory.

H. Appendices

39. Tables of elastic and dielectric constants. In this section the elastic and optical constants of some insulators are tabulated. The selection of crystals corresponds approximately to that of the phonon atlas (BILZ and KRESS, 1979). If possible, data at different temperatures are presented.

The knowledge of these macroscopic constants allows the dispersion curves for the crystal in question to be calculated if a "macroscopic" model is available (cf. Sect. 4). In the case of alkali halides, the breathing shell model (SCHRODER, 1966) seems to be a good approximation to such a model. Here it must be kept in mind that the measured elastic constants as given in this section are usually adiabatic or isothermal ones. For the calculation of phonons, however,

536 Appendices Sect. 39

"collision-free" constants are required which correspond to the long wavelength extrapolations of neutron scattering data. These data are, at present, not usually accurate enough for a precise determination of the elastic constants and of the difference between "collision-free" and other data, but there are indications that the static elastic constants have generally values a few percent lower than the collision-free constants (BUYERS, 1967; COWLEY, 1967). This difficulty does not exist for the optical constants. For convenience of the reader the references of the tables are given at the end of this section.

The reader who is interested in additional information is referred to the following collections of experimental data:

1. Landolt-Bornstein, 6th ed., Vol. II/8. Heidelberg: Springer 1962 2. Landolt-Bornstein, New series, Vols. III/iff. Heidelberg: Springer 1969ff. 3. G. SIMMONS and H. WANG: Single crystal elastic constants and calculated

agregate properties, 2nd ed. Cambridge: MIT Press 1971

Table 39 .• 1. Elastic (stiffness) constants (units: 1011 dyn.cm - 2)

Crystal Temperature C11 C12 C44

a) Rare-gas crystals

Ar 76.8 0.335 0.101 0.093 4.2 0.529 0.135 0.159

b) NaCI structure

LiH 300 6.531 1.485 4.501

LiD 300 6.626 1.462 4.553

LiF 300 11.049 4.435 6.368 300 11.373 4.759 6.368 300 11.111 4.208 6.296 293 11.397 4.767 6.364 293 11.302 4.766 6.369 293 11.120 4.200 6.280 293 10.730 3.964 6.357 200 11.742 4.257 6.417 100 12.295 4.213 6.466 80 12.366 4.202 6.471

4.2 12.445 4.264 6.471

LiCI 300 4.943 2.280 2.460 300 4.830 1.894 2.486 295 4.927 2.310 2.495 293 4.940 2.260 2.490 280 4.903 1.877 2.486 200 5.209 1.777 2.556 100 5.595 1.859 2.643 80 5.676 1.920 2.656

4.2 5.860 2.086 2.671

LiBr 300 3.937 1.870 1.930 293 3.940 1.880 1.910

* See end of this section

References *

2

2

3 4 5 6 7 8 9 5 5 5 5

3, 5 7 10 8 11 11 11 11 11

3 8



Crystal Temperature cll C12 C44 References *

LiI 300 2.850 1.400 1.350 12 293 2.850 1.400 1.350 8

NaF 300 9.700 2.430 2.810 3 300 9.700 2.380 2.822 4 300 9.630 2.459 2.794 10 300 9.710 2.430 2.800 8 293 9.695 2.363 2.820 14 280 9.749 2.444 2.807 10

80 10.710 2.312 2.897 10 4.0 10.850 2.290 2.899 10 0 11.039 2.242 2.947 14

NaCl 300 4.936 1.290 1.265 3 300 4.935 1.287 1.278 24 300 4.884 1.267 1.271 16 300 4.870 1.311 1.266 10 298 4.947 1.288 1.287 15 295 4.980 1.300 1.280 17 295 4.899 1.257 1.272 18 293 4.970 1.270 1.270 20 293 4.958 1.306 1.279 19 293 4.942 1.269 1.281 21 290 4.899 1.309 1.270 10 195 5.282 1.228 1.301 18 180 5.388 1.241 1.314 24 80 5.648 1.142 1.330 10 77.3 5.744 1.200 1.335 22 77 5.730 0.986 1.320 6

4.2 5.838 1.194 1.327 23 4.2 5.834 1.192 1.331 22 4.0 5.733 1.123 1.331 10,13

NaBr 300 4.012 1.090 0-.990 3 300 3.970 1.001 0.998 15 298 4.037 1.013 1.015 25 293 4.020 1.150 0.990 8 290 4.000 1.000 1.000 10

80 4.645 0.988 1.060 10 4.0 4.800 0.986 1.070 10,13

NaI 300 3.025 0.880 0.740 3,5 300 3.035 0.915 0.742 26 300 3.017 0.875 0.726 27 300 3.007 0.912 0.733 28 293 3.035 0.900 0.720 8 280 3.074 0.890 0.730 27 280 3.034 0.905 0.739 28 200 3.304 0.865 0.747 27 180 3.362 0.832 0.751 27

50 3.703 0.809 0.780 28 4.2 3.760 0.798 0.781 28

KF 300 6.560 1.460 1.250 3 300 6.480 1.600 1.252 29




Crystal Temperature cll c12 C44 References *

KF 298 6.485 1.427 1.281 25 295 6.490 1.520 1.232 10 293 6.580 1.490 1.280 8 280 6.547 1.591 1.256 29 180 6.943 1.567 1.277 29 20 7.571 1.538 1.293 29 4.2 7.585 1.473 1.293 29 4.2 7.570 1.350 1.336 10

KCl 300 4.078 0.690 0.633 3 300 4.069 0.711 0.631 15 300 3.997 0.698 0.666 16 300 4.032 0.660 0.629 30 298 4.069 0.711 0.631 15 298 4.035 0.651 0.633 31 295 4.050 0.698 0.630 18 293 3.990 0.680 0.630 8 293 3.980 0.620 0.625 20 293 4.090 0.704 0.627 21 280 4.095 0.650 0.632 30 280 4.085 0.608 0.634 32 195 4.050 0.658 0.064 18 180 4.394 0.600 0.646 30 180 4.458 0.612 0.651 32

80 4.697 0.550 0.657 30 80 4.807 0.580 0.664 32

4.0 4.832 0.540 0.663 6

KBr 300 3.476 0.570 0.507 3 300 3.468 0.580 0.507 15 300 3.498 0.562 0.538 16 300 3.419 0.520 0.508 33 293 3.420 0.560 0.500 8 293 3.460 0.580 0.505 20 293 3.500 0.620 0.506 34

4.0 4.170 0.580 0.505 6

KI 300 2.760 0.450 0.370 3,5 300 2.771 0.436 0.373 26 300 2.677 0.405 0.369 33 300 2.710 0.450 0.364 30 293 2.668 0.426 0.420 34 280 2.760 0.440 0.365 30 180 3.010 0.380 0.368 30 80 3.230 0.300 0.369 30 4.2 3.380 0.220 0.368 30,6

RbF 300 5.525 1.393 0.925 3 293 5.700 1.250 0.910 8

RbCl 300 3.643 0.615 0.465 3,5 300 3.658 0.615 0.475 35 300 3.653 0.645 0.478 11 300 3.646 0.647 0.468 10





RbCI 300 3.605 0.626 0.467 36 300 3.624 0.612 0.467 37 293 3.645 0.610 0.475 8 280 3.689 0.605 0.469 37 180 4.007 0.574 0.479 37 180 4.025 0.676 0.486 11 150 4.116 0.676 0.488 11 140 4.098 0.536 0.483 36 100 4.239 0.548 0.485 37 90 4.298 0.676 0.492 11

4.2 4.297 0.649 0.493 10 4.2 4.499 0.676 0.497 11

RbBr 300 3.157 0.495 0.380 3,5 300 3.163 0.467 0.384 35 300 3.150 0.493 0.384 6 300 3.107 0.515 0.376 10 300 3.170 0.420 0.388 38 300 3.155 0.493 0.380 33 300 3.152 0.500 0.380 37 293 3.185 0.480 0.385 8 290 3.135 0.512 0.377 10 280 3.204 0.494 0.382 37 180 3.488 0.459 0.390 37 140 3.597 0.444 0.393 37 80 3.725 0.474 0.404 10

4.0 3.863 0.474 0.408 10

RbI 300 2.583 0.370 0.278 3,5 300 2.573 0.377 0.279 35 300 2.540 0.407 0.276 6 300 2.550 0.340 0.277 10 300 2.556 0.382 0.278 37 300 2.560 0.310 0.287 38 293 2.585 0.375 0.281 8 290 2.575 0.340 0.278 10 260 2.652 0.364 0.280 37 180 2.850 0.335 0.284 37 120 2.988 0.314 0.287 37

4.0 3.21 0.360 0.292 10

AgCI 300 5.985 3.611 0.624 39 300 5.920 3.640 0.616 41 295 5.860 3.582 0.622 42 293 6.010 3.620 0.625 20 293 5.630 3.330 0.720 40 293 5.965 3.646 0.618 41 280 6.045 3.674 0.622 41 195 6.635 3.782 0.653 39 180 6.561 3.743 0.654 42 170 6.700 3.840 0.654 41

80 7.280 3.894 0.676 41 80 7.168 3.850 0.681 42





AgCI 0 7.391 3.907 0.694 42 0 7.590 3.908 0.689 41

AgBr 300 5.610 3.270 0.724 39 293 5.630 3.300 0.720 40 293 5.620 3.280 0.728 20 195 6.313 3.414 0.765 39

MgO 300 29.740 9.557 15.620 43 298 29.664 9.508 15.581 45 298 28.917 8.796 15.461 46 298 29.708 9.536 15.613 44 293 29.980 9.910 15.750 9 293 29.672 9.505 15.596 47 293 28.600 8.700 14.800 20 293 28.760 8.740 15.140 8 273 29.791 9.496 15.615 47 173 30.325 9.419 15.713 47 73 30.631 9.378 15.763 47

CaO 300 22.300 5.900 8.100 48 300 20.000 6.050 7.650 49 100 20.920 6.440 7.472 50

0 20.970 6.675 7.390 49

srO 300 17.300 4.500 5.600 48

MnO 298 22.300 12.000 7.900 51

CoO 303 26.170 14.500 8.320 52 293 25.911 14.871 8.286 52 273 19.673 16.729 8.070 52

PbTe 303.2 10.795 0.764 1.343 54 302 10.720 0.768 1.300 53 200 11.573 0.609 1.404 54 100 12.277 0.481 1.465 54

50 12.581 0.455 1.496 54 0 12.798 0.435 1.514 54

c) CsCl structure

CsCI 298 3.664 0.882 0.804 15 293 3.640 0.920 0.800 55 286 3.683 0.893 0.817 56

CsBr 300 3.000 0.780 0.756 38 300 3.056 0.776 0.743 57 300 3.072 0.800 0.725 58 300 3.091 0.841 0.747 59 298 3.063 0.807 0.750 15 295 3.097 0.903 0.750 6 293 3.100 0.840 0.750 55 286 3.683 0.893 0.817 56 280 3.071 0.818 0.759 57 280 3.118 0.915 0.761 6





CsBr 220 3.137 0.866 0.809 57 180 3.234 0.958 0.862 6

80 3.349 1.001 0.962 6 77 3.345 0.966 0.951 58 77 3.343 1.019 0.972 59 50 3.311 1.014 0.976 57 4.2 3.350 1.025 1.002 57 4.2 3.437 1.035 0.999 58

CsI 300 2.460 0.670 0.624 38 300 2.457 0.647 0.629 58 293 2.450 0.710 0.620 55 295 2.434 0.636 0.632 6 286 2.462 0.659 0.644 56 280 2.450 0.644 0.643 6 180 2.556 0.696 0.726 6 80 2.660 0.751 0.808 6 77 2.678 0.771 0.789 58 4.2 2.737 0.793 0.825 58 4.2 2.725 0.767 0.873 60

NH4CI 310 3.812 0.880 0.873 61 300 3.700 0.880 0.860 62 293 3.790 0.970 0.830 8 293 3.900 0.720 0.680 20 290 3.814 0.866 0.903 61 250 3.689 0.709 0.967 61 190 4.547 1.465 1.183 61 155 4.640 1.540 1.232 61

NH4BR 300 3.414 0.782 0.722 63 293 3.380 0.910 0.685 8 293 2.960 0.590 0.530 20 280 3.429 0.789 0.748 63 237 3.264 0.623 0.798 63

TICI 293 4.010 1.530 0.760 55

TlBr 300 3.734 1.498 0.748 64 298 3.760 1.458 0.757 65 293 3.780 1.480 0.756 55 270 3.807 1.533 0.774 64 250 3.870 1.496 0.810 65 200 4.000 1.544 0.870 65 180 4.056 1.646 0.867 64 100 4.280 1.618 0.995 65 90 4.314 1.766 0.963 64

4.2 4.469 1.881 1.007 64 0 4.339 1.660 1.079 65

d) Diamond structure

Diamond 300 110.000 33.000 44.000 20 300 107.600 12.500 57.580 66




Crystal Temperature Cll c12 C44 References *

Diamond 293 95.000 39.000 43.000 20 293 94.900 15.100 52.100 8

Si 298 16.578 6.394 7.962 67 293 16.570 6.390 7.960 20

77.2 16.772 6.498 8.036 67

Ge 298 13.160 5.090 6.690 20 298 12.835 4.823 6.666 69 298 12.890 4.830 6.710 68 173 13.050 4.900 6.788 68 73 13.150 4.948 6.840 68

e) Cubic ZnS structure

ZnS 302 9.760 5.900 4.510 70 298 10.462 6.534 4.613 71 293 10.000 6.500 3.400 72 293 10.320 6.475 4.620 73 273 10.350 6.485 4.630 73

93 10.550 6.540 4.680 73 77 10.711 6.664 4.655 71

ZnSe 298 8.096 4.881 4.405 71

ZnTe 298 7.134 4.078 3.115 71

GaP 300 14.120 6.253 7.047 74

GaAs 300 11.810 5.320 5.940 76 298 11.877 5.372 5.944 77 298 11.880 5.380 5.940 48 293 11.904 5.384 5.952 78

77.4 12.210 5.660 5.990 76

GaSb 298 8.850 4.040 4.330 48 298 8.839 4.033 4.316 79

AlSb 300 8.939 4.427 4.155 75

InSb 300 6.472 3.265 3.071 80 300 6.669 3.645 3.020 81 300 8.677 4.857 3.956 84 298 6.720 3.670 3.020 48 293 6.700 3.649 3.019 111 293 8.329 4.526 3.959 82,83 250 6.702 3.652 3.051 81 200 6.744 3.670 3.076 81 195 8.821 4.933 3.995 84 77 8.955 5.025 4.031 84 77.4 8.465 5.001 3.968 82 77.6 6.872 3.753 3.117 81 4.2 8.998 5.064 4.040 84 4.2 8.980 5.025 3.924 82 0 6.918 3.788 3.132 81

InP 293 10.220 5.760 4.600 85

SiC 293 35.230 14.040 23.290 86




f) Wurtzite st'ructure

Crystal Temperature cll c12 C44 References * C33 C55

BeO 298 46.060 12.650 8.848 88 49.160 14.770

293 47.000 16.800 11.900 87 49.400 15.300

CdS 300 8.432 5.212 4.638 90 9.397 1.489

298 8.431 5.208 4.567 91 9.183 1.458

298 9.068 5.809 5.094 71 9.380 1.504

293 8.160 4.950 4.790 8 8.080 1.430

293 8.100 4.900 4.800 89 8.000 1.430

CdSe 298 7.406 4.516 3.930 71 8.355 1.317

298 7.490 4.609 3.926 88 8.451 1.315

ZnO 298 20.970 12.110 10.510 92 21.090 4.247

g) Selen structure

Crystal Temperature cll c12 C1 3 References c14 C33 C44

Se 300 1.870 0.710 2.620 93 0.620 7.410 1.490

h) CaF2 structure

Crystal Temperature cll C12 C44 References

CaF2 300 16.400 5.300 3.370 94 298 16.420 4.398 3.370 95 293 16.400 4.500 3.380 8 293 16.357 4.401 3.392 96 280 16.400 5.400 3.390 94 195 16.750 4.516 3.490 95 180 16.700 5.500 3.487 94 80 17.200 5.300 3.562 94

4.2 17.400 5.600 3.593 94

BaF2 298 9.199 4.157 2.568 97 293 9.122 4.148 2.551 96 293 9.010 4.030 2.490 8 293 8.948 3.854 2.495 98 195 9.396 4.282 2.644 97




Crystal Temperature Cll C12 C44 References *

SrF2 300 12.350 4.305 3.128 99 280 12.390 4.342 3.144 99 180 12.620 4.521 3.219 99 80 12.820 4.695 3.291 99

4.2 12.870 4.748 3.308 99

SiMg2 300 12.100 2.200 4.640 100

SnMg2 300 8.240 2.080 3.660 101

U02 298 39.600 12.100 6.410 102

Th02 298 36.700 10.600 7.970 103

i) Rutile structure

Crystal Temperature Cll C12 C13 References C33 C44 C66

Ti02 298 27.143 17.796 14.957 104 48.395 12.443 19.477

298 27.140 17.800 14.960 120 48.400 12.440 19.480

293 26.600 17.330 13.620 105 46.990 12.390 18.860

293 27.300 17.600 14.900 106 48.400 12.500 19.400

200 27.710 18.460 120 20.300

100 28.420 19.240 120 21.620

20 28.850 19.690 120 22.710

4 28.860 19.700 120 22.720

MgF2 293 12.370 7.320 5.360 107 17.700 5.520 9.780

j) Ferroelectric crystals (perovskites,;:ABX 3)

Crystal Temperature Cll C12 C44 References

BaTi03 298 27.512 17.897 15.156 108 tetr. 16.486 5.435 11.312

cub. 423 17.278 8.196 10.823 108

SrTi03 293 34.817 10.064 45.455 109

KMnF3 300 12.037 3.715 2.761 110 280 11.999 3.734 2.758 110 200 10.789 4.340 2.665 110



Table 39.2. Static dielectric constant eo

Crystal 1.5K 90K 290K References *

LiF 8.50 8.58 9.01 112 8.50 9.00 113

9.035 114 8.09 8.29 8.81 116

LiCI 10.83 11.86 113

LiBr 11.95 13.23 113

NaF 4.73 4.80 5.05 112 4.73 5.08 113

5.07 114

NaCI 5.43 5.57 5.90 112 5.45 5.90 113

5.89 114

NaBr 5.78 5.92 6.28 112 5.78 6.27 113

6.397 114

NaI 6.61 6.80 7.28 112 6.62 7.28 113

KF 5.11 5.50 113

KCI 4.49 4.59 4.84 112, 113 4.81 114

KBr 4.52 4.64 4.90 112, 113 4.87 114

KI 4.66 4.81 5.10 112 4.68 5.09 113

RbF 5.99 6.48 113

RbCI 4.58 4.67 4.92 112 4.53 4.89 113

RbBr 4.51 4.60 4.86 112,113

RbI 4.55 4.65 4.91 112,113

AgCI 9.50 9.92 11.15 112, 113

AgBr 10.60 11.10 12.50 112 10.60 12.44 113

MgO 9.86 115 9.34 9.44 9.64 116

CaO 11.1 117 12.4 11.6 118

SrO 13.1 117 14.7 15.8 118

BaO 32.8 118

CsF 7.27 8.08 113

CsCI 6.68 6.95 113

CsBr 6.39 6.66 113

CsI 6.29 6.54 113

TICI 37.6 32.6 113

TlBr 35.1 30.4 113



Table 39.3. High frequency dielectric constant 8",

Crystal 2K 290K References *

LiF 1.93 1.93 113 LiCI 2.79 2.75 113 LiBr 3.22 3.16 113 LiI 3.89 3.80 113 NaF 1.75 1.74 113 NaCI 2.35 2.33 113 NaBr 2.64 2.60 113 NaI 3.08 3.01 113 KF 1.86 1.85 113 KCI 2.20 2.17 113 KBr 2.39 2.36 113 KI 2.68 2.65 113 RbF 1.94 1.93 113 RbCI 2.20 2.18 113 RbBr 2.36 2.34 113 RbI 2.61 2.58 113 AgCI 3.97 3.92 113 AgBr 4.68 4.62 113 MgO 2.956 115

3.01 116 CaO 3.33 117 srO 3.46 117 CsF 2.17 2.16 113 CsCI 2.67 2,63 113 CsBr 2.83 2.78 113 CsI 3.09 3.02 113 TICI 5.00 4.76 113 TIBr 5.64 5.34 113

Table 39.4. Infrared frequency roTO (units cm -1, 1012 S -1)

Crystal 2K 290K References *

[em-I] [S-I] [em-I] [S-I]

LiF 318 5.994 305 5.749 113 320 6.032 306 5.768 116

LiCI 221 4.165 203 3.826 113

LiBr 187 3.524 173 3.260 113

Lil 151.5 2.855 142 2.676 113

NaF 262 4.938 246.5 4.646 113

NaCI 178 3.355 164 3.091 113

NaBr 146 2.752 134 2.525 113

Nal 124 2.337 116 2.186 113

KF 201.5 3.798 194 3.656 113

KCI 151 2.846 142 2.676 113

KBr 123 2.318 114 2.148 113




Crystal 2K 290K References

[em- 1] [S-l] [em- 1] [S-1]

KI 109.5 2.064 102 1.922 113

RbF 163 3.072 158 2.978 113

RbCl 126 2.375 116.5 2.195 113

RbBr 94.5 1.781 87.5 1.649 113

RbI 81.5 1.536 75.5 1.423 113

AgCl 120 2.261 105.5 1.979 113

AgBr 91.5 1.724 79.5 1.498 113

MgO 408 7.691 401 7.559 116

CaO 295 5.561 117 302 5.693 311 5.862 118

SrO 231 4.354 117 224 4.222 229 4.317 118

BaO 146 2.752 118

CsF 134 2.525 127 2.393 113

CsCl 106.5 2.007 99.5 1.875 113

CsBr 78.5 1.479 73.5 1.385 113

CsI 65.5 1.234 62.0 1.168 113

TlCl 60.4 1.138 47.2 1.187 113

TlBr 63.0 0.889 47.9 0.902 113

References

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40. Tables of symmetry properties and selection rules for important crystal structures

Table 40.1. First Brillouin zones for important structures. [Notation after BOUCKAERT et al. (1936) and HERRING (1942)J

z

R I I A / I j:f T I .-

'/..1 X r;H.:::.--<>--- ---*'---+---

/' '-0.. Z Y "" E '

/ a Simple cubic

c Body-centered cubic (CsCl)

z

b Face-centered cubic (NaC!, diamond)

,,"

x r/

/

"

z

_-----..J...---_

d Hexagonal close-pocked (graphite)

y

Sect. 40 Tables of symmetry properties and selection rules 551

Table 40.2. Notations for the 10 irreducible representations for group Oh used by various authors. Representations for group ~ are given in the notation by WIGNER (W). EWK: EYRING et al. (1940); BSW: BoUCKAERT et al. (1936); vdLB: VONDER LAGE and BETHE (1947); HW: HOWARTH

and JONES (1952); K: KOSTER (1958); W: WIGNER (1936)

Dim. EWK BSW vdLB HJ K W ~ Basis cubic harmonics

A 1g r 1 ex TO r:+ I;.+ I;.+ 1

A l • 12, {J r 1 r.- r.- r,- xyz f 1 1 1 A lg 12 {J' Ii r.+ 12+ 12+ X4(yl _ zl) + y4(zl _ xl) + Z4(Xl _ y2)

2

A 1• I;., rx' Ik r:- r,- 12- x Y Z[X4(y2 _Z2) + y4(Z2 _ X2)+Z4(X2 _ y2)] 1 1

2 Eg I;. 2 l' r:1 r:+ I;.~ r,~ 3x2_r2, V3(y2_ Z2) d 3

2 E. I;. 2' 1" r" r:- I;.2 r'2 xyz(4x2-r2), V3xYZ(yl_z2) 3

3 Tlg 125' e I'.Jl r,+ 4 12~ 12~ yz, zx, xy

3 Tlu I;. 5 b r" r.- I;.5 125 x, y, Z 5 3 T1g I;. 5' b' ro r.+ I;.~ r,~ YZ(yl_Zl), ZX(Zl_Xl), xy(xl_yl) 5 3 Tl • 125 e' r l r.- 125 125 X(yl _ Zl), Y(Zl _ Xl), Z(Xl _ yl)

f 4

Table 40.3. Vibrational symmetries of common crystals in the notation of Koster (1958), Bracketed representations are degenerate (from HAVES and LoNDON, 1978)

Crystal Class Vibrational mode symmetries Other examples

Acoustic Raman- Raman· active optic' inactive optic'

Sodium nitrite mm2 C2v r, +12+~ 3I;. +312 NaNOl +1;+2&

Calcium tungstate 4jm C4h I;.- 2I;.+ +512+ 4r,- + 3r,- CaM 00, SrMo04

CaW04 +(1;-+~-) +5(1;++~+) +4(1;- +14-)

Barium titanate 4mm C4v I;. +r5 3I;.+r3+ 4 !.;. BaTi03

Strontium titanate 4jmmm D4h 12- +Fs- r,+ +21;+ 2rl+ + r,- + 312-SrTi03 +~++3Fs+ +~-+515-

Rutile TiOl 4jmmm D4h 12- +Fs- I;.;-+r3 + 12+ +12- +21;- MnFl' FeFl , CoFl +~++Fs+ +3r5-

rx-Quartz SiOl 32 D3 r l +1; 4r, + 8& 412

Bismuth Bi 3m D3d 12-+1;- I;.+ +r3+ As, Sb

Lithium iodate 6 C6 I;. +(r5+16) 4r,+5(rl +r3) 5~ LiI03 +4(~+16)

Wurtzite ZnS 6mm C6v r,+r5 I;. +~+216 2r3 ZnO, CdS, BeO

Hexagonal close 6jmmm D6h 12- + Fs- r.+ 6

r:+ 3 Be, Mg, Cd, Zn

packed

Zinc blende ZnS 43m ~ Fs ~ GaAs, GaP

Cubic perovskite m3m Oh r4 3~-+Fs- BaTi03, SrTi03

CaTi03

a Infrared active modes are underlined



Crystal Class Vibrational mode symmetries Other examples

Acoustic Raman- Raman-active optic· inactive optic·

Fluorite CaF2 m3m Oh 1:-4 1:+ s 1:-...±.. SrF2, BaF2, AuAl2

Diamond C m3m °h 14- 1:+ s Si, Ge

Rocksalt NaCI m3m °h 1:-4 1:---±- KBr, NaI

Caesium chloride m3m Oh 14- 14-CsCI

Simple cubic, m3m Oh 14-bee, fcc

• Infrared active modes are underlined

Table 40.4. Raman-active vibrational symmetries and Raman tensors for the crystal symmetry classes (from LoUDON, 1964)

System Class Raman tensors

Monoclinic

C b:) (, » 2 C2 A(y) B(x,z)

m C3 A'(x, z) A"(y)

21m C2h Ag Bg

Orthorhombic

(" b J (: H ') ( : ) 222 D2 A Bi (z) B2(y) B3(X) mm2 C2v A i (z) A2 Bi (x) B2(y) mmm D2h Ag Big B2g B3g

Trigonal (" )( , ") ( ,-, -f) a b : -; f =; -: e

3 C3 A(z) E(x) E(y) 3 C~i Ag Eg Eg

(" " bH -d b -, -') 32 D3 Ai E(x) E(y) 3m C3v A i (z) E(y) E(-x)

3m D3d A ig Eg Eg



System Class Raman tensors

Tetragonal (" 'J (: -: ) () ) C, -~) 4 C4 A(z) B E(x) E(y)

4 S4 A B(z) E(x) E(-y)

4/m C4h A. B. E. E.

(,'J (' -c H' )( , ) ( ,) e

4mm C4v Al(z) Bl Bl E(x) E(y)

422 D4 Al Bl Bl E(-y) E(x) 42m Dld Al Bl Bl(z) E(y) E(x)

4/mmm D4h A l• Bl• Bl • E. E.

Hexagonal (" , J C, :) C, -:) (; -~ ) (-~ =; ) 6 C6 A(z) El(X) El(y) El El 6 C3h N En En E'(x) E'(y)

6/m C6h Ag E lg E lg Elg Elg

(' , J ( c ,) C -'H' ) (' -d )

622 D6 Al El(x) El(y) El El 6mm C6v Al(z) El(y) El(-x) El El 6m2 D3h N 1

En En E'(x) E'(y)

6/mmm D6h A lg Elg E lg Elg Elg

Cubic (' J (b b J (V3b V3b J ( :H d) (: ) a

23 T A E E F(x) F(y) F(z)

m3 T. Ag Eg Eg Fg Fg Fg 432 ° Al E E Fl Fl Fl 43m Td Al E E Fl(x) Fl(y) Fl(z)

m3m 0. Al• E. E. Fl. F. Flg


Table 40.5. Two-phonon processes in diamond structure (from BIRMAN, 1974)

Point Two phonons Type Critical-point Activity' index

Overtones

r [r<25 + )](2) 20 P3 R

*X [*X(4)] (2) 2TO Po R [*X(I)](2) 2L (lW)+P3(1)) R [*X(3)](2) 2TA P, +F2 R

*L [*L(3 - )](2) 2TO Pz R [* L(1 + )](2) 2LO Pz R [* IJ2 - )](2) 2LA 1't R [*I.!3+ )](2) 2TA 1't R

*1: [*1:(2)](2) 2T01 Pz+1't(l) R [*1:(1)](2) 2T02 Pz(l)+1't R [*1:(3)](2) 2LO 1't R [* 1:(3)](2) 2TA1 P3 R

*Q [*Q(2)](2) 2T02 P, R [*Q(2)](2) 2TA1 P2 R

*8 [*8(2)](2) 2T02 Pz R

Combinations

*X *X(4)@*X(I) T01,2+LO,LA J;(1) R;IR +(Pz(l) or F,(l))

*X(4)@*X(3) TO 1, 2+ TAl, 2 1't +F2 R;IR *X(I)@*X(3) L+TA1,2 Po(1)+F2(1) R;IR

*L *L(3-)@*L(I+) TO+LO Pz IR *L(3-)@*IJ2-) TO+LA 1't R *IP-)@*I.!3+) TO+TA 1't IR *I.!'+)@*I.!2-) LO+LA J; IR *I.!'+)@*I.!3+) LO+TA Po+Pz R *I.!2-)@*I.!3+) LA+TA 1't IR

*W *W(2)@*W(2) TO(l)+ TO (2) 1't R *W(I)@*W(l) LO+LA P3 R *W(2)@*W(2) TA(1)+TA(2) P2 R

*LI *LI(1)@*LI(5) LO+TA1,2 J; R;IR

*1: *1:(1)@*1:(3) T01,2+LO 1't R;IR *1:(l)@*1:(3) T01,2+TA1 J; R;IR *1:(3)@*1:(3) LO+TA1 P2 R;IR

*Q T01,2+TA2 Pz R;IR LO+LA 1't R;IR LO+TA1 Pz R;IR(15R)

*8] TA1+TA2 1't R;IR

*8II LO+LA Pz R;IR LO+TA1 1't R;IR LO+TA2 Pz R;IR TA1+TA2 Pz R;IR

a R means Raman active (at least one allowed representation present; IR means infra-red active, r(15-) present). From J.L. BIRMAN: Phys. Rev. 131, 1492 (1963)


Table 40.6. Two-phonon processes in rocksalt structure (after BIRMAN, 1974)

Point Two phonons Type Activity'

Overtones

r [r(15 - )J(2) 20 R

*X [*X(4-)J(2) 2LA,2LO R [*X(5-)J(2) 2TA,2TO R

*L [* L!' + lJ(2) 2LA R [*L!2-)J(2) 2LO R [*L(3+)J(2) 2TA R [* L(3 - )J(2) 2TO R

*1: [*1:(1)J(2) 2TO(1) R [* 1:(2)J(2) 2TO(2) R [* 1:(3)J(2) 2LO R [*1:(3)J(2) 2TA(1) R [* 1:(4)J(2) 2TA(2) R

Combinations

*X X(4-)<8>*X(5-) L+T R

*L L!'+)<8>L(2-) LA+LO IR L!' +)<8>L!3+) TA+LA R L!1+)<8>L(3-) TO+LA IR L!2-)<8>L!3+) TA+LO IR L!2-)<8>L(3-) TO+LO R L(3+)<8>L(3+) TA+TO IR

*1: All combinations except 1:(3) + 1:(4) IR

• R = Raman active, I R = infrared active

556

Species

[r(15)](2)

[*X(5)] (2)

[*X(1)](2) [*X(3)] (2)

[*lP)](2) [*LJ ' )](2)

[*w(m)l2) m= 1, 2, 3, 4

*X(5) 0*X(1) 0*X(3) 0*X(5)

*X(3) 0*X(I)

*LJ3) 0*LJ')

* LJ3) 0* LJ3) *LJ ' ) 0*LJ' )

*W(1)0*W(1) *W(2)0*W(2) *W(3)0*W(3) *W(4)0*W(4) *W(1)0*W(2) *W(1)0*W(3) *W(1)0*W(4) *W(2)0*W(3) *W(2)0*W(4) *W(3)0*W(4)

Appendices

Table 40.7. Two-phonon processes in zincblende (from BIRMAN, 1974)

Activityb

D;R

D;R

~ } D(2); R D;R

D; R} D;R D;R D;R

D;R

D;R D;R

R R R R D;R D;R D;R D;R D;R D;R

Overtones

Type

20(0

2 TO (X) and 2TA(X)

2LO(X) and 2LA(X)

2 TO(L) and 2TA(L) 2LO(L) and 2LA(L)

20,(W); 20 2(W); 20 3(W); 2A ,(W); 2A 2(W); 2A3(W)

Combinations

TO(X) +LO(X); TO(X)+LA(X); TA(X) + LO(X); TA(X) + LA(X) TO(X) + T A(X) LO(X) + LA(X)

TO(L) + LO(L); TO(L) + LA(L); TA(L) + LO(L); TA(L) + LA(L) TO(L)+TA(L) LO(L) + LA(L)

Sect. 40

a In listing these combinations it is not possible to be more specific because of ambiguity of assignments at * W However, in this list of combinations, it is to be noted that the two phonons participating, even if of the same symmetry, must arise from different branches

b In this table we use the symbol D(n) to signify that the representation L(15) occurs n times in the reduction and therefore there is an "n-fold" infrared dipole activity.

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ed. by H.M. Crosswhite, H.W. Moos (Wiley, New York) p. 409 Yin, M.T., Cohen, M.L. (1980): Phys. Rev. Lett. 45,1004 Yonezawa, F. (1964): Prog. Theor. Phys. Kyoto 31,357 Yonezawa, F. (1968): Prog. Theor. Phys. Kyoto 40,734 Yonezawa, F., Matsubara, T. (1966a): Prog. Theor. Phys. Kyoto 35,357 Yonezawa, F., Matsubara, T. (1966b): Prog. Theor. Phys. Kyoto 35, 759 Yuasa, T., Mitsuishi, A., Komiya, H., Ibuki, S. (1970): Japan. J. App!. Phys. 9,1421 Zakrzhevkii, V.I., Pikhtin, A.N., Yas'kov, D.A. (1971): Fiz. Tverd. Tela 13, 2635 [English trans!.:

SOY. Phys.-Solid State 13, 2210 (1972)) Zavt, G.S. (1963): Fiz. Tverd. Tela 5,1086 [English trans!.: SOY. Phys.-Solid State 5, 792 (1963)) Zavt, G.S. (1977): Phys. Status Solidi B 80,399 Zavt, G.S., Kristofel, N.N., Khizhnyakov, V.V. (1965): Fiz. Tverd. Tela 7, 2444 [English trans!.:

SOY. Phys.-Solid State 7, 1973 (1966)) Zeks, B., Shukla, G.c., Biinc, R. (1971): Phys. Rev. B 3, 2306 Zeyher, R. (1971): Phys. Status Solidi B 48, 711 Zeyher, R. (1975): Phys. Rev. Lett. 35, 174 Zeyher, R. (1978): In Lattice Dynamics, ed. by M. Balkanski (Flammarion Sciences, Paris) p. 17 Zeyher, R., Bilz, H. (1968): In Localized Excitations in Solids, ed. by R.F. Wallis (Plenum, New

York) p. 767 Zeyher, R., Bilz, H. (1969): Phys. Status Solidi 31,157 Zeyher, R., Bilz, H., Cardona, M. (1976): Solid State Commum. 19, 57 Zigone, M., Beserman, R., Fair, H.D. (1976): In Light Scattering in Solids, ed. by M. Balkanski,

R.C.C. Leite, S.P.S. Porto (Flammarion Sciences, Paris) p. 597 Zigone, M., Kunc, K., Plumelle, P., Vandevyver, M. (1978): In Lattice Dynamics, ed. by M. Balkanski

(Flammarion Sciences, Paris) p. 405 Zinken, A., Buchenau, U., Fenzl, H.J., Schober, H.R. (1977): Solid State Commun. 22, 693 Zittartz, J. (1974a): Solid State Commun. 14,51 Zittartz, J. (1974b): Z. Phys. 267, 243 Zubarev, D.N. (1960): Usp. Fiz. Nauk 71, 71 [Englisch trans!.: SOY. Phys.-Usp. 3, 320 (1960))

Subject Index

Absorption, infrared (see infrared absorption) Acoustic sum rule 75 Adiabatic approximation (or Born-Oppen

heimer approximation) 14-16,112,349, 363, 367, 378

Alkali halides (see also substances) 22, 25, 27, 31, 51, 68, 71, 162, 199, 234, 280, 342, 343, 379

Amorphous systems 227, 384 Anharmonicity 6, 77, 377, 381, 406 -, Coulomb 180 -, cubic 171, 183 -, defects 284, 286, 376, 378, 406, 446, 463 -, Green function 138, 376, 416, 480ff, 527 -, potential 142, 166, 187, 446, 478 -, quartic etc. 172 -, -, overlap 223 Antiresonance 320, 324, 371

Berreman technique 197, 391 Boltzmann equation, phonon 105 Bond polarizabilities 60, 70, 157, 257, 262 Bond-charge model 46, 47, 50, 59, 72, 76, 187,

213, 216, 367 Bond-dipole model 47 Bond-orbital model 72, 73 Born approximation, first 80 Born model 22 Born-Huang conditions 38, 39 Born-Mayer potential 22, 37, 52, 132, 379, 425 Born-Oppenheimer approximation (see

adiabatic approximation) Bound states, two-phonon 190, 191, 254, 475 Boundary conditions, periodic 497 Breathing shell model 29, 30, 42, 46, 235, 245,

291, 292, 318, 345 Brillouin scattering 147, 246 Brillouin zones, table 550 Brout's sum rule 215

Cauchy relation 30, 38, 40-43, 52 Charge neutrality 10, 12, 15, 75 Charges -, apparent 332 -, atomic (effective) 73 -, bond charge 29, 44, 75, 76 -, Born (transverse effective) 52, 182

-, Callen (longitudinal effective) 187 -, defects 265, 288, 333, 334, 338, 365 -, effective 61, 65-67, 69, 358, 359 -, -, defects 334, 335, 373 -, electronic (effective) 70, 154 -, ionic effective 61, 102 -, longitudinal 66 -, Lundqvist 52 -, overlap 28, 68 -, polarization 61, 67, 74, 188 -, renormalization 76 -, shell model 373 -, static 25, 29, 30,42, 61, 67, 69 -, Szigeti (effective) 27, 67-69, 74, 188,

215 -, transfer 63, 332, 338 -, transverse (effective) 52, 66, 69, 72, 75, 94,

187, 215 Clausius-Mosotti relation 63, 67, 374 Cluster expansion 491, 492, 498, 500, 511,

531, 533 Coherent potential approximation 382, 388,

391, 393 Cohesive energy 22, 25, 59 Compressibility 24, 51, 194,464 Condon approximation 349. Correlation functions 14, 479, 480 - -, current-current 82 - -, density-density 82 - -, static 101 - -, time autocorrelation 81 Coulomb anharmonicity 166, 180, 182,407 - forces 21,23,28,37,52, 180, 363,

379, 425 Covalency 44, 48, 59, 72 Critical points 18, 141, 389 - -, analysis 167, 209 - -, defect-activated 289-294, 331, 389 Crystals, covalent, ionic etc. (see substances) Cumulants 475, 492

Damping function 140, 159, 166, 168, 169, 179,181-183,189,196,201,418,431

- -, density-of-states approximation 169 - -, quartic anharmonicity 175,184 Debye relaxation 70 Debye-Waller factor (see Huang-Rhys factor)

594 Subject Index

Defects (see also critical points, gap, local, resonance modes) 262

-, bond-charge model 367 -, charge changes 265, 288, 333, 365, 388 -, concentration effects (see also coherent po-

tential approximation, virtual crystal approximation) 288, 289, 336, 351, 382, 394, 411, 451

-, - -, self-consistent approximation 388 -, - -, single site scattering approximation

386, 391, 393, 451 -, effective charge 334, 335, 373 -, effective mass 326, 327, 330, 340,421,429 -, elastic constants, local 450, 459, 462, 464 -, - -, compliance tensor 458 -, electric-field effects 451 -, force-constant changes 264, 293, 300, 323,

334, 336, 344, 347, 362, 370-372, 440, 467 -, Green-function 296, 315-318, 324, 366, 384 -, hindered rotations 351 -, image strain 454 -, impurity space 310, 311, 323, 324, 326, 330,

336, 382, 412, 452 -, infrared absorption 209, 266, 273, 289-295,

331-340, 371 -, interstitials 286, 351, 352 -, isoelectronic defects 264 -, isotope effects 264, 268, 289-291, 295, 383,

428, 434 -, lattice distortion (see lattice distortion) -, local modes (see local modes) -, many-defect perturbation matrix 382 -, molecular 263, 280, 283, 347, 351, 355 -, molecular model 288, 295, 369 -, off-center 280, 282, 406 -, pairs 288, 347, 383, 384, 405 -, pressure induced effects (see -, stress) -, quasi-localized modes 272 -, Raman spectra, defect-induced 266, 292,

341, 377, 384, 389, 393 -, - scattering, ionic 341, 381 -, resonance modes (see resonance modes) -, resonance Raman scattering 347 -, rigid-ion models 332 -, shell model 332, 334, 358 -, strain, local (see lattice distortion) -, stress 449, 454, 466 -, substitutional 305 -, symmetry considerations 302, 334, 362 -, thermal conductivity 391 -, V-center (see substances) Deformability 29, 31, 33, 34, 38, 40, 42, 60 -, breathing 46, 245 -, shell models 33, 171, 365 Deformation, (see also lattice distortion) -, homogeneous 129, 132,452, 453, 459, 497 -, potential 125

Deformation dipole model 28, 29, 49, 74, 292, 345,346

Density matrix 97,479 Density-of-states 17 -, critical points 289 -, defective crystals 266, 289, 293, 328 Diagrams, non-linear dipole moments 186,

377 -, defect scattering 386-388 -, free energy 532 -, Raman scattering 153, 380-381 -, self-energy 413, 502, 518 -, susceptibility 374· -, vertex part 512 Dielectric constant 4, 24, 65, 68, 93, 158, 165,

217, 535 - -, electrons 86 - -, Hartree approximation 257 - -, hydrostatic pressure 248 - susceptibility 331, 374, 393 - tensor 247 Difference bands, two-phonon 162 -, -, three-phonon 162 Dipole - approximation 26,61,84,85,92,96, 115 - forces 28, 45, 57 - gauge 86, 88, 92 - matrix elements 87, 111 - model 26, 44, 49, 53, 56, 57 - -, deformation 28, 29, 49, 74, 292, 345 - operator 122 - sum rule 71, 87, 88 - sums, two-dimensional 472 Dipole moments, auto-correlation function

100 - -, non-linear 8, 59, 98, 135, 156, 186, 188,

208, 210, 263, 360, 376, 438 - -, renormalized 141, 377 - -, second-order 135, 200, 212, 360, 377,

439 - - static 97 Dirac's identity 100 Dislocations 289 Dispersion oscillators (see also Reststrahlen os

cillator) 7, 21, 69, 95, 134, 136, 138-140, 159, 163

- -, coupling between 142 - -, damping function 172 - -, line widths 191, 195 - -, volume-dependence 195 Dispersion relations 51, 86, 220 Distortions (see lattice distortions) Drude part, infrared absorption 218 Dynamical matrix 17 Dynamic form factor 81, 82 Dyson equation 138, 316, 386, 408, 474, 500,

503, 504, 507, 527

Subject Index 595

Edge modes 471 Eigenvalue equations 477 - treatment 323, 340, 411 Einsteinoscillator 2, 296, 312, 318, 320, 326,

414, 416, 426, 463, 465 Elastic constants 42, 44, 449, 535 - -, Born relation 44 - -, defects 450, 459, 462, 464 Electric field - -, effective 62, 66 - -, internal 61, 66, 361, 365, 367 - -, local 64, 70, 75 - -, local-field correction 249, 258 - -, Lorentz field 60, 62, 66 - -, Lorentz-Lorenz formula 75, 248 - -, macroscopic 4, 23, 25, 62, 63, 70 - -, static external 127, 132, 147, 451 Electronegativity 70 Electron-electron interaction 26, 51 Electronic polarizability 28, 46, 74, 378 - propagator 378, 381 - susceptibility 87, 88, 133, 143, 154, 341 Electron-phonon interaction, adiabatic 7, 15,

26, 32, 34, 51, 58, 67, 75, 110, 155, 208, 223, 249, 349, 378

- -, covalent crystals 213 - -, microscopic 151 - -, model treatment 152,378 - -, multiple expansion 29, 86, 215 - -, non-adiabatic 113 - -, unitary transformation 114 Electron-photon interaction 79 Energy, cohesive 22 Equation of motion 16 - -, normal coordinates 495 - -, perturbed lattice 300 - -, unitary transformation 303 Equilibrium condition 24, 38, 40 - configuration 15 - Green function 492 - positions 298, 497, 498, 531, 534, 535 Ergodicity 106 Ewald technique 23 - Theta-function transformation 130 Exchange-charge model 53

F-center 347, 361, 448 Fermi, golden rule 80 - resonance 190 Ferroelectricity 219, 282, 359 - soft modes 60 Finite crystals, continuum theory 467 - -, semi-infinite 467 - -, slabs 467 - strains 448 Fluctuation-dissipation theorem 78, 101,486 Foldy-Wouthuysen representation 82

Force constant 16, 17, 364 - -, defect-induced changes (see defects) - -, effective 361, 364 - -, formal 39, 41, 44, 364 Free energy 13, 14, 21, 103, 491, 494, 504, 530 - -, perturbation theory 532 - enthalpy 491 Fresnel's equation 95 Frohlich interaction 79 - mode 468 f-sum rule 88 Functional derivatives 475, 487, 489, 492 - - method 140, 487 - -, lineare responce 487

Gap modes 267, 288, 314, 318, 320, 322, 324, 400

Gauge invariance 78, 82, 86 -, dipole 86, 88, 92 -, multipole 86 -, radiation 83, 91 -, transverse 83, 86 Green function (phonon) - -, anharmonic 376,416 - -, eigenvalue treatment 323 - -, equilibrium 492 - -, four-time 510, 513 - -, generalized retarded 491 - -, harmonic 481 - -, Hartree 118, 525 - -, Hartree-Fock 510 - -, integrated 510 - -, many-time 511 - -, non-equilibrium 487 - -, one-phonon 137, 483 - -, perturbed (see defects) - -, Raman scattering 116,342 - -, real 315, 317 - -, resonance mode 326, 327 - -, retarded 98, 482, 484, 502, 506, 524 - -, time-ordered 475, 480 - -, two-phonon 135, 138, 518, 525 Group theory 18,303,311 Griineisen parameter 194

Hamiltonian 12 -, harmonic 134 -, interaction 84, 88, 96, 117 -, lattice 126, 127, 131,476 -, non-relativistic 91 -, radiation field 117 Harmonic oscillators, renormalized 479 - -, thermodynamic expectation value 482 Hartree-Fock approximation, Green function

510 - -, calculations 21, 232 - -, polarizabilities 74

596 Subject Index

Heider-London approximation (HLA) 52, 54 Hellmann-Feynman theorem 58, 113, 530 Historical survey 2 Huang-Rhys factor 420, 423

Impurities (see defects, substances) Infrared absorption 4, 7, 77, 95, 114, 149, 164,

186, 213, 331, 358, 373, 409 - -, absorbed energy 99 - -, band mode 290, 293 - -, bibliography 203 - -, concentration effects 382, 394 - -, crystals, covalent, ionic, etc. (see sub-

stances) - -, defect-induced (see also critical points, gap

modes, local modes, resonance modes) 209, 266, 273, 289-295, 331-340, 371, 439

- -, diagrams 186 - -, dispersion oscillators (see dispersion oscil-

lator) - -, Drude part 218 - -, electric-field-induced 451 - -, high-frequency 178 - -, integrated intensity 333, 341, 375, 419 - -, isotopes 289 - -, low-frequency 183 - -, low-symmetry 224 - -, mixed crystals 349 - -, model theory 148, 358 - -, multi-phonon 20,162, 178, 180,418,420,

434 - -, non-linear dipole moments 8, 59, 98, 135,

156, 186, 188,208,210,263, 360, 376, 438

- -, pressure-dependence 195,405,439,450, 463

- -, Reststrahlen band (see Reststrahlen oscil-lator, dispersion oscillator)

- -, small particles 469 - -, spectra of crystals 157 ff. - -, sum rule 321 - -, temperature dependence (see also self-ener-

gy) 195 Ionic crystals 15, 21, 61, 157, 161 - -, lattice energy 21 - plasma frequency 23 - polarizabilities 60, 190 Ionicity 59, 71, 72 Isotope effects (see defects) Ivey relation 405

Jellium 13

Kellermann model 25,27, 30, 50, 165 Kramers-Kronig relations 5, 89, 101, 184, 193,

319,320,324,325,417,486 Kubo formula 96, 101

-, linear-response theory 490 Kurosawa formula 206, 221

Lattice dynamics 2 - -, adiabatic condition 34 -, dipole moment 132 -, distortions, defect-induced 297-300, 310,

337, 365, 405, 440, 445, 450, 451, 460 -, energy 22 -, Hamiltonian 126, 127, 131, 476 -, polarizability 143 -, potential 16, 114, 126 - statics 297 - susceptibility 133, 141 Ledermann theorem 315 Librations 284 Lifshitz formalism 8, 296, 302, 309, 311,

363 Light scattering 79 - -, electrons 90 - -, inelastic (see Raman scattering) - -, polaritons 123 - -, transition rate 118 Local modes 267, 287, 288, 295, 314, 318, 320,

322, 324, 330, 391, 398, 400, 412, 421, 435, 436,466

- -, crystals with impurities (see substances) - -, intensity 327, 422 - -, isotope effects 267, 268, 295, 296 - -, line-width (see self-energy) - -, shift under pressure 463 - -, sideband 283, 286, 371, 423, 441, 444 - -, U center 296, 368, 389,415,428, 432,

437,461 Lorentz field 60, 62, 66 - -Lorenz formula 75, 248 Lyddane-Sachs-Teller relation 24, 66, 70, 76,

94, 95, 102, 204, 206, 395

Madelung sums 13, 22 Magnetic dipole interactions 85 Many-body forces 5, 22, 38, 40, 41 Mass defects 312, 318, 321, 322, 340, 461 Metallicity 59, 70, 72, 73 Microscopic theory 51, 57,473 - -, phonons 74 Mixed crystals 263, 394 - -, mode behaviour 394, 399-404 - -, strain 405, 460 Model theory, dynamics of perturbed lattices

358 - -, infrared absorption 148, 358 - -, Raman scattering 148, 358 Molecular crystals 43 - defects 263, 283, 347, 351, 355 - model 288, 295, 369 Morse relations 19

Subject Index 597

Multi-phonon processes 20,162, 178, 180, 418, 420, 434

Neutron scattering 78, 390 Newton's integral 13 Normal coordinates 17,476,495 n-point functions 492,498, 507, 514

Optical constants 4, 5, 27, 75, 89, 102, 171, 536

Oscillator strength, infrared 66, 162, 399 Overlap effects 25, 30, 43 - polarization 36, 54 - shell model 36, 37, 50,260 - theory 29, 42, 52, 74 Overton absorption 347,418,434,436

Partition function 13, 487 Pauli's principle 10, 15 Peierls-Bogoljubov inequality 13 Penn's formula 70 - gap 250 Perturbation matrix 302, 365 - -, many defects 382 - theory, free energy 532 - -, thermodynamic 488 Perturbed crystals (see also defects) 262 Phillips' model 70, 72 Phonon 9 -, Boltzmann equation 105 -, creation operators 126, 478 -, destruction operators 126, 478 -, dynamics 21, 494 -, eigenvectors 58, 301, 325, 476 -, field operators 126 -, final-states interaction 190 -, frequency, anharmonic 199 -, Green function (see also Green function) 5 -, lifetime 505 -, - phonon interaction 113, 531 -, - photon interaction 6, 8, 25, 117 -, polaritons 93 -, pseudo-harmonic 7, 20, 21, 40, 163, 165 -, quasi-harmonic 21 -, renormalized harmonic 500, 501, 505, 527,

533 -, self-consistent 532 -, self-energy (see self-energy) -, transport 510, 523 Photo-elastic (elasto-optic) coefficients 147,

246-249, 251, 255, 332, 342 photon self energy 118 - spectral density 119 Photon-electron interactions 8 Photon-exciton interaction 6 Photon-phonon interactions 6, 8, 25, 117 Piezo-electric crystals 108, 448 Piezo-optical coefficients 247

Placzek approximation 109, 116 Pockels coefficients (see photo-elastic

coefficients) Point group, matrix equations into different

irreducible representations 303 - symmetry 408 Polaritons 8, 92, 95, 123, 124, 468, 470 Polarity of crystals 73 Polarizability (see also lattice polarizability,

Raman scattering) 62, 63 -, bond charge 150 -, defects 311, 339, 358-360, 363, 366, 371 -, electronic 28, 46, 74 -, expansion theory 125 -, inter-ionic 150,232,237,260,262 -, intra-ionic, quartic 241, 243 -, ionic 60, 190 -, oxygen 60 -, theory 109, 115 Potential, lattice 478 -, -, central 24, 40, 171 -, -, cubic 175 -, -, harmonic 15, 34, 40, 477, 500, 505, 533 -, -, non-central 25, 38, 40 -, -, quartic 178 Pressure effects 147, 194, 195,405,439,445,

450,463 Projection operator 20 Propagators (see also Green function) 5, 6, 474 Pseudopotential 46, 54, 68, 71-73 Pyroelectric crystals 108

Quadrupolar deformability 28, 32, 33 - sum rwe 89 Quadrupole forces 29

Raman scattering 4, 5,77, 108, 114, 143,230, 231, 234, 409

- -, active oscillator 145, 339, 342 - -, anti-Stokes processes 79, 120, 144 - -, anti-symmetric part 123, 146 - -, bond charge 156 - -, bond polarizability 156,253 - -, critical point analysis 238 - -, cross section 5 - -, - - absolute 80 - -, - -, differential 81, 119, 341 - -, - -, power 80 - -, - -, quantum 80 - - crystals (see substances) - -, defect-induced 266, 292, 341, 377, 384,

389,393 - -, diagrams 153, 380-381 - -, dynamic form factor 116 - -, external pressure 147 - -, field-induced 147,252 - -, first-order 144, 251

598 Subject Index

Raman scattering (continued) - -, Green function theory 116 - -, higher-order 144, 380 - - integrated scattered intensity 343 - -, ionic Raman effect 108, 341, 381 - -, model theory 148, 358 - -, w4-law 120 - -, perturbation theory 111 - -, perturbed crystals (see - -, defect-induced) - -, polariton picture 109, 113 - -, polarizabilities 60, 146 - -, polarizability theory 109, 115 - -, quantum theory 109 - -, resonance 124, 146, 230, 347, 381, 404 - -, scattering - -, - angle 78 - -, - efficiency 80 - -, - geometry 79 - -, - volume 79 - -, selection rules 233 - -, shell model 153, 377 - -, Stokes processes 79, 120, 144 - -, symmetry tensors 145 - -, tensor 120, 122, 342, 552 Rayleigh modes 473 - scattering 90, 143, 342 reflectivity (see infrared absorption) Refractive index 78, 175, 332 Relativistic CTP-invariance 106 Resonance modes (see also gap modes, local

modes) 271,275, 288, 293, 320, 326, 337, 371, 406, 412, 416, 422, 424, 463

- -, intensity 326,337,417,419 - -, shift (see also pressure) 424, 442 - -, width 326, 417, 431, 442, 434, 435 Reststrahlen bands 159, 162, 396-405 - oscillators (see also dispersion oscillator) 7,

98, 135, 161, 335, 338, 342 Rigid-ion model 21,24,25, 52, 76 - -, defects 332 Rigid-overlap approximation 52

Screening, long-range forces 215 Selection rules 18, 20, 266, 406, 408 Self-energy 6, 7, 21, 109, 138-140, 163, 193,

380,407,413,446,486, 506, 507, 524 -, anharmonic (defects) 327, 446 -, diagrams 502 -, effective (defects) 410 -, harmonic 336, 386, 404 -, one-phonon 155 -, photon 118 -, self-consistency 503 -, vertex corrections 516 Shell model 17, 26, 28, 30, 45, 57, 59, 74, 291,

299, 311, 345, 358, 364 - -, breathing 29, 30,42,291, 292, 345

- -, defects 332, 334, 358, 442 - -, deformable 33, 171,365 - -, dipole moments 376 - -, Lifshitz formalism 363 - -, non-linear 149, 376 - -, overlap 36, 37, 50,260 - -, valence-overlap 260 Shift function (see also self-energy) 418, 444 Sidebands 283, 286, 371, 418, 423, 434, 437,

441,444 Small particles, infrared absorption 469 Soft modes 219, 359 Spectral representations 483 Spectral-density function 100,484 Stability conditions 14 - of solids 10 Stark effect 451 Strain (see lattice distortions) -, finite 448 -, homogeneous 452, 453, 459, 460, 463 -, infinitesimal 448 - in mixed crystals 405 Stress (see pressure) - optical coefficients 247 Substances (for further informations about

crystals with defects cf. tables 20.1-20.5): AgBr 239 AgBr:Li+ 422,436 AgCl 32, 189, 199, 239 AgCl:OH- 283 alkali chlorides 230 alkali fluorides 238 alkali halides, defects 269,272, 286, 331 - -, F center 347 - -, infrared spectra 158, 159, 163, 178, 180 - -, lattice vibrations 30ff., 69 - -, mass defect 340 - -, Raman spectra 230, 237 - -, U center 264, 267, 269, 289, 295, 347,

368, 371,405,415,418,422,436,443,461 alkali hydrides 240 alkaline earth halides 204, 269, 389 alkaline earth oxides 231, 240 amorphous systems 227, 384 BaF 2 204, 389 BaF2 :H- 422,434,435 BaTi03 219 C (diamond) 209, 252 CaF2 203,204 CaF2:Eu3+ 361 CaF2:H- 422 CaO 216,241 CaO:Cu2+ 341 carotene (fJ-) 350 Ca1 _ xSrxF2 396,397,404 CdF2:In3+ 347 CdO 224

Subject Index 599

CdS 50, 224, 225 CdS:Be 437 CdS:Mn 448 CdSe: Be2 + 437 CdS1_xSex 398,399,401,405 CdTe:Be2+ 422,437 cesium halides 199, 238 cha1cogenides 30, 200, 244, 245 compounds, II-VI 25, 69, 216, 258 -, III-V 25, 69, 258 -, mixed-valence 43 crystals, covalent 48, 73, 208, 214, 215, 252 -, cubic, Raman oscillator 145, 203 -, ferroelectric 207, 359 -, ionic 230,247, 331 -, oxides 199 -, group-IV 267,288, 331, 338, 342, 398 -, low-symmetry, infrared spectra 224 -, mixed 263, 394, 460 -, mixed ionic and covalent 214 CsBr 161,162,200,292,368 CsCI 284, 307, 332, 347 CsI 292 CsI:K+ 293 CsI:T1+ 422 diamond (C) 44,48, 51,73,208, 209, 252,

257, 375, 554 EuO 244 EuS 245 FeO 244 ferroelectrics 207, 359 GaAs 49, 58, 214, 215, 217, 258, 288 GaASO.94P 0.06 404 Ga1 _)nxSb 402,403 GaP 49, 216, 470 GaP:B 437 Ge, germanium 44,208,209,212,217,228,

252,257,288 -, IX-Ge (amorph) 228 Ge1 - xSix 398 glasses 227 H- (V-center) (see U-center) H;- (V 1 center, KBr) 286, 287 H1-xDx 290 hydrides 234 InAs 216 InP 1 _ xAsx 398 InSb 216, 218 KBr 161,173,175,178,184,191,195,234,

237, 285, 292, 294, 345, 372, 443 KBr:D- 434 KBr:H- (V-center) 434,439,440,441,444 KBr: H;- (V 1) 286, 287 KBr:Li+ 390,422,430 KBr: 6Li+ and 7Li+ 429 KBr: Tl + 344, 345 KBr)1_x 406

KCl 161, 179, 184, 230, 267, 281, 292, 293, 294, 345, 347

KCl:CN- 351 KC1:D- 428,433 KCl, F centers 348 KCl:H- 370,418,422,423,428,433,465,

466 KCl: 6Li 281 KCl:Na+ 288 KCl:Na+ (pair) 422 KCl:Tl+ 291 KC11 _ xBrx 395,405 (KCl)1_iNH4Cl)x 390 KI 160, 292, 300, 301, 350, 443 KI:Ag+ 422,428,431,437 KI:Cl- 288 KI:H;- 286 KI:Na+ 288 KI:Rb+ 288 KI:Tl+ 347 K 1 _ xRb) 405 KTa03 219, 245 LiBr 69 LiCI 390 LiD 232,240,241 LiF 59,69, 161, 168ff., 179ff., 189, 196,

248, 290, 438, 469, 470 LiF:H- 269,438 LiH 232,240 LiH1_xDx 404 6Li/Li1_xF 290 MgO 60, 69, 74, 216, 232, 239, 244, 283,

472 MgO:OH- 283 molecular crystals 43 NaCl 160, 161, 174, 184, 195,230,237,273,

290, 292, 293, 300, 305, 336ff., 347 NaCl:Ag+ 288, 422 NaCl:Cu+ 273,422,431,437,438,448 NaC1:F- 288,337 NaCl:H- 269 NaCl:KF 288 Na35Cl/7CI1_x 290 NaI 28, 312, 318, 327 NaI:Cl- 422,431 NaI: 35Cl and 37Cl 429 NaI:H- 269,422 NH4 Cl1 _ xBrx 404 NO; molecule 351 OH- molecule 281-284 oxides 199, 231, 234 oxygen 0 2 -, polarizability 60, 150, 233 PbSe 202 PbTe 359 perovskites 37, 307, 219, 245 pnictides 245, 258 potassium halides 347,406

600 Subject Index

Substances (continued) (for further informations about crystals with defects cf. tables 20.1-20.5): rare-earth halides 271, 396, 436 rare-gas crystals 331 RbCI 32, 230, 237 RbF 69,471 rocksalt structures 22, 302, 397, 555 Se, selenium 214, 224, 226 Si, silicon 45, 208, 211, 252 ff. -, a-Si (amorph) 228 silver halides 33, 198, 240, 436 Sn, grey tin 44, 48, 208 SnTe 359 SrxBa1_xF 2 396 SrF2 204ff. SrF2 :H- 422 SrO 241 SrTi03 60, 219ff., 245 sulfur 214 Te, tellurium 214, 224 Ti02 :OH- 284 TlBr 51 U-centers 264, 267, 269, 288, 289, 295ff.,

347, 367ff., 389, 405, 415, 418, 422, 428, 432, 436ff., 443, 461

wurtzite structures 49, 224, 264, 271, 396ff. zinc cha1cogenides 219, 259 ZnO 224, 259, 261 ZnS, zinc blende 49, 55, 220, 259ff., 264,

271, 288, 307, 347, 396 ZnS:Mn 448 ZnSe 50 ZnS1_,,sex 405 ZnTe 260

Sum rrues 50, 71, 75, 86, 86--88, 90, 101,215, 321

Superconvergence theorem 89, 103 Surfaces 39, 289, 467 - modes 468, 471 Susceptibility (see also infrared absorption) 65,

78, 101 -, adiabatic 104, 105 -, anharmonic 137

-, electronic 87, 88, 133, 143, 154, 341 -, isothermal 104, 105 -, Kubo 101 Symmetries, common crystals 8, 440, 551 -,phonons 18 -, set of critical points 19 Szigeti (effective) charge 27, 67-69, 74, 188,

215

T matrix 315, 323 - resonance 316, 326 Thermal conductivity, perturbed crystals 391 Thermal expansion 406,427,445,450,452,

466 Thermodynamic limit 13 - potentials 491 - stability 12, 14 Thomas-Fermi theory 10, 11 Thomson cross section 91 Time-reversal invariance 476 Transition-polarizability 90, 109, 115 Tunneling 285 -, absorption 282 -, motion 280

U-centers (see substances)

Vacancies 264 Valence forces 44,48,49 - -, bond-bending potential, Keating 47 - -, bond-stretching forces 33 Valency 69 Van-der-Waals interaction 14,22 Vertex part, cubic 508 - -, diagrammatic 512 - -, functions 510 - -, higher-order 508 - -, integral equations 509, 512 - renormalization 509 Vibrational symmetries 552 Vibronic processes 8, 361, 367 Virtual-crystal approximation 382, 390, 399,

400 Volume modes 471

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Emphasis is placed on the following fields:

Solid-State Physics Semiconductor Physics: H.J.Queisser, MPI Stuttgart Amorphous Semiconductors: M.H.Brodsky, IBM Yorktown Heights Magnetism (Materials, Phenomena): H.P.J. Wijn, Philips Eindhoven Metals and Alloys, Solid-State Electron Microscopy: S.Amelinckx, Mol Positron Annihilation: P. Hautojiirvi, Espoo Solid-State lonics W. Weppner, MPI Stuttgart

Surface Science Surface Analysis: H.Ibach, KFA Jillich Surface Physics: D.Mills, UC Irvine Chemisorption: R.Gomer, U. Chicago

Surface Engineering Ion Implantation and Sputtering: H. H. Andersen, U. Aarhus Laser Annealing: G. Eckhardt, flughes Malibu Integrated Optics, Fiber Optics, Acoustic Surface Waves: R. Ulrich, TV Hamburg

Coordinating Editor: H. K. V. Lotsch, Heidelberg

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Light Scattering in Solids I Introductory Concepts

Editor: M. Cardona 2nd corrected and updated edition. 1983. 111 figures. XV, 363 pages. (Topics in Applied Physics, Volume 8) ISBN 3-540-11913-2

Contents: M. Cardona: Introduction. - A. Pinc:zuk, E. Burstein: Fundamentals of Inelastic Light Scattering in Semiconductors and Insulators. - R. M. Martin, L. M. Falicov: Resonant Raman Scattering. - M. V.Klein: Electronic Raman Scattering. - M. H. Brodsky: Raman Scattering in Amorphous Semiconductors. - A. S. Pine: Brillouin Scattering in Semiconductors. - Y.-R. Shen: Stimulated Raman Scattering. - Overview. - Additional References with Titles. - Subject Index. - Contents of Light Scattering in Solids II, ill and IV.

Light Scattering in Solids II Basic Concepts and Instrumentation

Editors: M. Cardona, G. Giintherodt 1982. 88 figures. XIII, 251 pages (Topics in Applied Physics, Volume 50) ISBN 3-540-11380-0 Contents: M. Cardona, G. Guntherodt: Introduction. -M. Cardona: Resonance Phenomena. - R. K Chang, M. B. Long: Optical Multichannel Detection. - H. Vogt: Coherent and Hyper-Raman Techniques. - Subject Index.

Light Scattering in Solids III Recent Results Editors: M. Cardona, G. Giintherodt 1982. 128 figures. XI, 281 pages (Topics in Applied Physics, Volume 51) ISBN 3-540-11513-7 Contents: M. Cardona, G. Guntherodt: Introduction. -M. S. Dresselhaus, G. Dresselhaus: Light Scattering in Graphite Intercalation Compounds. - D. J Lockwood: Light Scattering from Electronic and Magnetic Excitations in Transition-Metal Halides. - W. Hayes: Light Scattering by Superionic Conductors. - M. V. Klein: Raman Studies of Phonon Anomalies in Transition-Metal Compounds. -JR. Sandercock: Trends in Brillouin Scattering: Studies of Opaque Materials, Supported Films, and Central Modes. -C. Weisbuch, R. G. Ulbrich: Resonant Light Scattering Mediated by Excitonic Polaritons in Semiconductors. - Subject Index.

Documents

Light and Matter Id / Licht und Materie Id